![15
[ Chapter One ]
Symbols and Language in
the Early Modern Period
The alphabet is really now superfluous
for in this sign all men can find salvation.
—Goethe, Faust, Part II (trans. Atkins)
Idols and Hieroglyphs
In the scientific circles of the seventeenth century, words had a bad reputa-
tion. In the 1623 version of his book The Advancement of Learning, Francis
Bacon warned against what he called the “idols of the market”—
the “vul-
gar” notions that, in everyday speech, tend to “insinuate themselves into
the understanding” by means of words.1 As a protection against “the seduc-
ing incantation of names,” he tentatively suggests definitions and “terms of
art,” but even these are not enough; truly preventing words from “doing
violence to the understanding,” he states, will require “a new and deeper
remedy.”2 At almost exactly the same time, there was an explosion of new
mathematical symbols.3 In the mid-
1500s, algebra often took the form of
words, with even equations, which we now think of as made out of sym-
bols, appearing in knotty prose. By the mid-1600s, this logorrhea had given
way to compact symbolic expressions like ax + b = c. Although Bacon
himself had little interest in mathematics, scholars have long noted an al-
liance between these new symbols and his followers’ hostility toward lan-
guage.4 Algebraic notation, brought into something like its modern form
by Thomas Harriot and René Descartes in the early decades of the 1600s,
came to be associated with a philosophical ideal of clarity, and numerous
thinkers, G.W. Leibniz among them, envisioned developing analogous
symbols for all manner of subjects.
This chapter gives an overview of the symbolic methods that existed be-](https://image.slidesharecdn.com/24359676-250524020316-f2e7efd4/85/Language-And-The-Rise-Of-The-Algorithm-Jeffrey-M-Binder-26-320.jpg)
![20 ‹ Chapter One
Chinese language.22 Bacon explains this quality as a departure from the
Aristotelian model: the characters “express, not their letters [i.e., speech
sounds] or words, but things and notions; insomuch, that numerous na-
tions, though of quite different languages, yet, agreeing in the use of these
characters, hold correspondence by writing.”23 As a result, “a book written
in such characters may be read and interpreted by each nation in its own
respective language.”24
Although Bacon does not discuss mathematical symbols in the passage,
it later became routine to cite numerals and algebraic symbols as examples
of real characters.25 For instance, the mathematician John Wallis wrote
that, like Chinese characters, algebraic signs “so little need the interven-
tion of Words to make known their meaning, that, when different persons
come to express, in Words, the sense of those Characters, they will as lit-
tle agree upon the same Words, though all express the same sense, as two
Translators of one and the same Book into another Language.”26 Robert
Hooke made a similar comparison with regard to “Arithmetical Figures.”27
Since it does not align with how modern linguistics understands writing,
a number of modern scholars have dismissed the idea of a real character
as mistaken or absurd; Jaap Maat goes so far as to call the idea a “myth.”28
But the idea is not wholly senseless when applied to mathematical sym-
bols. Calling these symbols real characters amounts to claiming that they
express universal ideas that are accessible to all people, regardless of what
words one chooses when pronouncing them—
that the English word mi-
nus and the French word moins really share a common core of meaning
by means of which the minus sign can be used to communicate the idea of
subtraction clearly across languages.29
This is not to say that the real-
character idea still holds water. The idea
depended on a faith that the human mind was divinely constructed to mir-
ror the world, which precludes any serious recognition of cultural diver-
sity. Not everyone thought this way in the seventeenth century. But nu-
merals and algebraic symbols really are, in a sense, more like an alternative
to the alphabet than a language. As the next section shows, seventeenth-
century textbooks taught numerals in a way that emphasized physical pen
skills and speaking numbers aloud. This pedagogy had more in common
with learning to read and write (or with learning a shorthand like Bright’s
“characterie”) than with learning a second language. There was also a dif-
ference in regard to gender: vernacular tongues were largely learned from
women—
from mothers and nurses—
whereas writing and mathematics
alike were both typically learned from male teachers.30 Examining how
numerals were taught sheds light on why early modern philosophers such
as Leibniz put so much stock in the power of symbols—
and on how the](https://image.slidesharecdn.com/24359676-250524020316-f2e7efd4/85/Language-And-The-Rise-Of-The-Algorithm-Jeffrey-M-Binder-31-320.jpg)
![28 ‹ Chapter One
solving six types of equation; he then discusses mercantile calculations,
techniques for computing the areas of geometric figures, and calculations
involving inheritance. This final section is indicative of the uses to which
algebra was put in al-
Khwārizmī’s context. The Abbasid Caliphate had
complex rules for inheritance that required fulfilling certain equation-
like
conditions; al-
Khwārizmī’s discussion of these issues takes up almost half
the book.61
Like the surviving translations of al-
Khwārizmī’s arithmetic, the alge-
bra is oriented toward teaching the reader a set of practical methods. In
contrast to the direct instructions of the algorism, he explains his alge-
braic methods through specific problems, such as “half of a square and five
roots are equal to twenty-
eight dirhems [a unit of currency].”62 That is, in
modern notation, x2
/2 + 5x = 28. His solution for this problem begins as
follows: “Your first business must be to complete your square, so that it
amounts to one whole square. This you effect by doubling it. Therefore
double it, and double also that which is added to it, as well as what is equal
to it. Then you have a square and ten roots, equal to fifty-
six dirhems.”63
What he is instructing us to do here—
doubling the coefficients—
is specific
to this instance of the problem: if the first number is something other than
a half, one has to multiply by some other number. Yet this example is meant
to stand in for a broader class of problems. “Proceed in this manner,” he
concludes the section, “whenever you meet with squares and roots that
are equal to simple numbers: for it will always answer.”64 Al-Khwārizmī’s
compiling of these instructions marks the beginning of what Victor J. Katz
and Karen Hunger Parshall call the “algorithmic stage” in the history of
algebra—
a stage in which algebra consisted primarily of procedures for
solving particular classes of problem.65
Like algorism, algebra was long regarded as a practical matter in Eu-
rope. Al-
Khwārizmī himself would have had no strong reason to classify
his work as either practical or theoretical—
Islam, as it was interpreted by
the Abbasids, encouraged a continuum between secular and holy knowl-
edge.66 European universities had comparatively rigid hierarchies of pres-
tige, and algebra did not self-
evidently deserve the status of a learned dis-
cipline. Unlike arithmetic and geometry, algebra had no place among the
seven liberal arts; its diffusion in Europe was largely due to merchants such
as Leonardo of Pisa (Fibonacci), who encountered Arabic mathematics
while accompanying his father on a trading expedition to North Africa and
described algebraic methods in his 1202 Book of Calculation.67 During the
Renaissance, the Italian mercantile academies known as “abacus schools”
taught algebra alongside a range of calculating practices.68 While algebra
was sometimes taught at universities, its academic status long remained](https://image.slidesharecdn.com/24359676-250524020316-f2e7efd4/85/Language-And-The-Rise-Of-The-Algorithm-Jeffrey-M-Binder-39-320.jpg)
![32 ‹ Chapter One
The best-
known instance of this issue appears in the work of the Italian
polymath Gerolamo Cardano. In his 1545 book The Great Art, or, the Rules
of Algebra, Cardano described a complete solution for cubic equations—
that is, in modern notation, equations of the form ax3
+ bx2
+ cx + d = 0.
The desire for such a solution had been long-
standing; the Persian mathe-
matician Omar Khayyam (best known as the presumed author of the poetic
cycle the Rubaiyat) had analyzed cubic equations in the eleventh century,
but neither he nor his immediate followers could find an algorithmic solu-
tion.92 Cardano’s solution was attended by a well-
known scandal that gives
a taste of what life as a mathematician was like in sixteenth-
century Italy.
Cardano learned part of the solution, as he acknowledges, from an unpub-
lished poem shared by an acquaintance known as Tartaglia (which means
“the stammerer”). Tartaglia swore him to secrecy about this result, and yet
Cardano published it anyway. Cardano had an excuse. Tartaglia, he had dis-
covered, was not the first to discover the result; another mathematician
named Scipione del Ferro had discovered it over a decade before. Cardano
took this to mean it was fair game to publish. Yet he failed to acknowledge
the oath, which led to a bitter dispute culminating in a Renaissance alterna-
tive to a duel: a public mathematics contest between Tartaglia and one of
Cardano’s students, Ludovico Ferrari, who bested the stammerer and put
the matter to an end.
The solution Cardano assembled breaks the problem down into more
specific cases such as x3
+ ax = b and x3
= ax2
+ b. This division into cases
is necessary because, like al-
Khwārizmī, Cardano does not allow equa-
tions to have negative coefficients; it is thus impossible to combine all cu-
bic equations into one general form. His “rules” take the form of knotty
prose with the occasional use of an abbreviation, such as ℞ for “root.” To
solve x3
+ ax = b, for instance, one follows these instructions:
Cube one-
third the coefficient of the number of things, add it to the
square of one-
half the constant of the equation; & take the square root
of the whole. You will duplicate this, and to one of the two you add the
one-
half of a number you have already squared and from the other you
subtract the same. You will then have a binomium and its apotome. Then,
subtracting the cube ℞ of the apotome from the cube ℞ of the binomium,
the remainder [or] that which is left is the value of the thing.93
The historian of mathematics Helena M. Pycior characterizes these pro-
cedures as “prose algorithms,” and they can readily be interpreted as al-
gorithms in the modern sense.94 Unlike al-
Khwārizmī and Stifel, Cardano](https://image.slidesharecdn.com/24359676-250524020316-f2e7efd4/85/Language-And-The-Rise-Of-The-Algorithm-Jeffrey-M-Binder-43-320.jpg)
![34 ‹ Chapter One
bitrary.”99 These “uncertain” symbols seemed to many thinkers, paradoxi-
cally, to achieve a level of clarity words could not match—
a development
that almost immediately inspired dreams of a universal method.
From Numbers to Letters
In today’s primary schools, the transition from arithmetic to algebra is
marked by the sudden appearance of letters. After years of learning about
fixed values such as 4 + 5, the student is suddenly confronted with expres-
sions such as 4 + x and must learn how to reason about a value that is not
yet known. The need for some way of referring to unknown quantities is
essential to algebra, but the use of letters to fill this office is a relatively re-
cent development.100 Modern algebraic notation is based on an idea that
would later become central to the design of programming languages—
the
use of arbitrary symbols to represent values that are left unspecified for
the purpose of generalizing a procedure. This use of symbols has become
deeply entrenched in modern algorithmic thinking, but it was not there in
the work of al-
Khwārizmī himself, and it came at the cost of placing com-
putational procedures in a fraught relation to meaning.
Early iterations of Arabic algebra most commonly represented un-
known quantities with words. Medieval Arabic writers referred to the un-
known as shay (ءْ َ
ش), meaning thing; in Latin, this became res, which is the
word Cardano uses. But long before Cardano’s time, other ways of express-
ing unknowns had appeared. Robert of Chester’s c. 1145 Latin translation
of the Al-jabr includes a condensed summary of the “rules” (regul[a]e) of
the art that uses symbols not in the Arabic original. In his critical edition,
Barabas B. Hughes approximates these symbols as ø, ϑ, and ʒ. For instance:
“When ø is equal to ϑ and ʒ, ø and ϑ must be divided by ʒ, ϑ halved, the half
drawn into itself, the product added to the number. The radix of the aggre-
gated whole minus half ϑ reveals what is sought.”101 The letters correspond
to coefficients attached to terms of specific degrees: ø is a constant, ϑ the
unknown, ʒ the square of the unknown. In modern terms, then, the equa-
tion is ʒx2
+ ϑx = ø.
Yet placing Chester’s symbols in the company of modern notation like
that is misleading. His ø, ϑ, and ʒ are not interchangeable tokens like the as
and bs of modern algebra; they function more like common nouns in that
they have, to borrow a pair of terms from Gottlob Frege, both sense and
reference.102 In Frege’s terms, reference is what is singled out: the reference
of the phrase the liar, for instance, is the person being called a liar. Sense is
the semantic freight the words carry: in this case, all it is to be a liar. Ches-
ter’s symbols employ both types of meaning, at once referring to numeri-](https://image.slidesharecdn.com/24359676-250524020316-f2e7efd4/85/Language-And-The-Rise-Of-The-Algorithm-Jeffrey-M-Binder-45-320.jpg)
![of a plough in its seven bright stars; and was probably so
denominated by the earliest astronomers, before the introduction of
the Zabian idolatry, as a celestial symbol of agriculture? The thick
cudgel corresponds to the brazen mace of Homer. And it is highly
probable that the Assyrian Nimrod, or Hindu Bala, was also the
prototype of the Grecian Hercules, with his club and lion’s skin.
Nimrod is said to have been a mighty hunter before the Lord;”
which the Jerusalem paraphrast interprets of a sinful hunting after
the sons of men to turn them off from the true religion. But it may
as well be taken in a more literal sense, for hunting of wild beasts;
inasmuch as the circumstance of his being a mighty hunter is
mentioned with great propriety to introduce the account of his
setting up his kingdom; the exercise of hunting being looked upon in
ancient times as a means of acquiring the rudiments of war; for
which reason, the principal heroes of Heathen antiquity, as Theseus,
Nestor, &c, were, as Xenophon tells us, bred up to hunting. Beside, it
may be supposed, that by this practice Nimrod drew together a
great company of robust young men to attend him in his sport, and
by that means increased his power. And by destroying the wild
beasts, which, in the comparatively defenceless state of society in
those early ages, were no doubt very dangerous enemies, he might,
perhaps, render himself farther popular; thereby engaging numbers
to join with him, and to promote his chief design of subduing men,
and making himself master of many nations.
NINEVEH. This capital of the Assyrian empire could boast of the
remotest antiquity. Tacitus styles it, Vetustissima sedes Assyriæ;”
[the most ancient seat of Assyria;] and Scripture informs us that
Nimrod, after he had built Babel, in the land of Shinar, invaded
Assyria, where he built Nineveh, and several other cities, Genesis x,
11. Its name denotes the habitation of Nin,” which seems to have
been the proper name of that rebel,” as Nimrod signifies. And it is
uniformly styled by Herodotus, Xenophon, Diodorus, Lucian, &c, Ἡ
Νίνος, the city of Ninus.” And the village of Nunia, opposite Mosul, in
its name, and the tradition of the natives, ascertains the site of the
ancient city, which was near the castle of Arbela, according to
Tacitus, so celebrated for the decisive victory of Alexander the Great](https://image.slidesharecdn.com/24359676-250524020316-f2e7efd4/85/Language-And-The-Rise-Of-The-Algorithm-Jeffrey-M-Binder-68-320.jpg)
![ejected from their livings by the Act of Uniformity in 1662; the
number of whom, according to Dr. Calamy, was nearly two
thousand; and to the laity who adhered to them. The celebrated Mr.
Locke says, Bartholomew-day (the day fixed by the Act of
Uniformity) was fatal to our church and religion, by throwing out a
very great number of worthy, learned, pious, and orthodox divines,
who could not come up to this and other things in that act. And it is
worth your knowledge, that so great was the zeal in carrying on this
church affair, and so blind was the obedience required, that if you
compare the time of passing the act with the time allowed for the
clergy to subscribe the book of Common Prayer thereby established,
you shall plainly find, it could not be printed and distributed, so as
one man in forty could have seen and read the book before they did
so perfectly assent and consent thereto.”
By this act, the clergy were required to subscribe, ex animo,
[sincerely,] their assent and consent to all and every thing contained
in the book of Common Prayer,” which had never before been
insisted on, so rigidly as to deprive them of their livings and
livelihood. Several other acts were passed about this time, very
oppressive both to the clergy and laity. In the preceding year 1661,
the Corporation Act incapacitated all persons from offices of trust
and honour in a corporation, who did not receive the sacrament in
the established church. The Conventicle Act, in 1663 and 1670,
forbade the attendance at conventicles; that is, at places of worship
other than the establishment, where more than five adults were
present beside the resident family; and that under penalties of fine
and imprisonment by the sentence of magistrates without a jury.
The Oxford Act of 1665 banished nonconforming ministers five miles
from any corporate town sending members to parliament, and
prohibited them from keeping or teaching schools. The Test Act of
the same year required all persons, accepting any office under
government, to receive the sacrament in the established church.
Such were the dreadful consequences of this intolerant spirit, that
it is supposed that near eight thousand died in prison in the reign of
Charles II. It is said that Mr. Jeremiah White had carefully collected a
list of those who had suffered between Charles II. and the](https://image.slidesharecdn.com/24359676-250524020316-f2e7efd4/85/Language-And-The-Rise-Of-The-Algorithm-Jeffrey-M-Binder-76-320.jpg)
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- 6. Language and the Rise of the Algorithm
- 8. . . . Language and the Rise of the Algorithm . . . Jeffrey M. Binder The University of Chicago Press Chicago and London
- 9. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 2022 by The University of Chicago All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission, except in the case of brief quotations in critical articles and reviews. For more information, contact the University of Chicago Press, 1427 E. 60th St., Chicago, IL 60637. Published 2022 Printed in the United States of America 31 30 29 28 27 26 25 24 23 22 1 2 3 4 5 ISBN-13: 978-0-226-82253-2 (cloth) ISBN-13: 978-0-226-82254-9 (e-book) DOI: https://doi.org/10.7208/chicago/9780226822549.001.0001 Library of Congress Control Number: 202201812 This paper meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).
- 10. Contents Introduction 1 Chapter One Symbols and Language in the Early Modern Period 15 Chapter Two The Matter Out of Which Thought Is Formed 50 Chapter Three Symbols and the Enlightened Mind 86 Chapter Four Language without Things 123 Chapter Five Mass Produced Software Components 163 Coda The Age of Arbitrariness 204 Acknowledgments 227 Notes 229 Bibliography 275 Index 311
- 12. 1 Introduction /* * If the new process paused because it was * swapped out, set the stack level to the last call * to savu(u_ssav). This means that the return * which is executed immediately after the call to aretu * actually returns from the last routine which did * the savu. * * You are not expected to understand this. */ if(rp->p_flag&SSWAP) { rp- >p_flag =& ~SSWAP; aretu(u.u_ssav); } — Lions’ Commentary on UNIX 6th Edition, with Source Code The Compromise In May 2020, as much of the world was focused on the COVID- 19 pan- demic and as racial justice protests took place across the United States, a technical development sparked excitement and fear in narrower circles. A computer program called GPT- 3, developed by the OpenAI company, produced some of the best computer- generated imitations of human writ- ing yet seen: fake news articles that were, according to the authors, able to fool human readers nearly half the time, and poems in the style of Wal- lace Stevens.1 The program is based on a statistical model that does one thing: given a sequence of words, it tries to predict what word will come
- 13. 2 ‹ Introduction next. The model was trained on more than 570 gigabytes of compressed text scraped from the internet in addition to the contents of Wikipedia and a large number of books.2 The system’s creators describe it as a “task- agnostic” learner— that is, a machine learning model that can perform a wide range of cognitive tasks without having to be fine- tuned for any par- ticular one.3 This new approach to artificial intelligence (AI) aspires to transform the practice of computer programming: instead of designing an algorithm to solve a given problem, one tells the machine its goal in En glish, and it works out (one hopes) the correct answer. From a humanistic standpoint, a striking aspect of this claim is how it locates knowledge in language. GPT- 3’s input and output consist of text, and it is trained on nothing but text; it has no experience, even in the loos- est notional sense, of anything whatsoever.4 Yet its apparent capabilities are not limited to such language- oriented tasks as rewriting paragraphs in different styles; to the extent that it really is a multitask learner, it unites the functions of writing aid, programmable calculator, and search engine. Skeptically viewed, the machine is acting like a parrot, saying things it cannot understand. But the idea that a language model can form the basis for a universal method could also suggest something like a deconstructive insight: that learning language cannot be distinguished from learning to think, that there is no limit to the sorts of cognitive operations that go into choosing words. If we are to believe the researchers— which we certainly should not do uncritically— then natural language is the essential ingredi- ent needed to create the elusive artificial general intelligence (AGI). The rise of large language models such as GPT- 3 has unsettled the cat- egories in which people have long understood the relation of computation to language. Computers are often described as symbol- manipulating ma- chines; they work by rearranging electrically represented ones and zeros through mechanical rules that do not depend on the symbols’ meanings. GPT- 3 has rekindled a long- standing philosophical debate over whether such a machine can really be said to understand a language.5 But even be- fore this development, computers have seldom been used as purely un- interpreted symbol manipulators. In modern interfaces, screens are fes- tooned with words— save, submit, like— that serve to mediate between computational logic and the social conventions by which people com- municate. Engineers have long treated the communicational elements of computer systems as superficial ornaments when compared to the data structures and algorithms that form the real core of a computer program. Language models such as GPT- 3 have blurred the lines. Since these sys- tems depend, through and through, on data about people’s linguistic prac-
- 14. Introduction › 3 tices, they make it harder than ever to judge where algorithm ends and lan- guage begins. The term algorithm, as it is used in computer science, is notoriously easier to illustrate than to define. While the word has recently become as- sociated with machine learning, textbooks typically explain algorithms, quite simply, as precisely defined procedures for solving problems. These procedures often take the form of sequences of steps, as in the following algorithm for finding the length of the longest sentence in a book: Write the number 0 on scrap paper For each sentence in the book, repeat the following: Count the number of words in the sentence If the result is greater than the number on the scrap paper: Replace the number on the scrap paper with the result Although similar instructions occur in a wide range of contexts— a typi- cal example is cooking recipes— calling a procedure an algorithm evokes a more specific set of disciplinary practices. Programming languages pro- vide a way of describing procedures with the extreme precision demanded by machines. (To make the foregoing procedure a true algorithm, we would have to clarify what words and sentences are— not a straightforward matter.) Computational complexity theory provides methods for gauging and improving the efficiency of these procedures. More broadly, algorith- mic thinking (in the expansive sense of thinking about algorithms) invites abstraction.6 The technical theory of algorithms encourages the develop- ment of general solutions that can be reused for different purposes and in different contexts; the procedures are thought of as mathematical entities that exist apart from the complexities of the languages in which they are described and the concrete situations in which they are used. This book is about how this form of abstraction came into being. It fo- cuses on one thread in the prehistory of algorithms: the use of symbols in numerical calculation, algebra, calculus, logic, and, eventually, computer science. Standard programming languages such as Python and R draw (among other sources) on the symbolic notations of algebra and logic as ways of precisely defining operations. Yet these notations, like program- ming languages, have long combined computation with another function that is harder to reduce to mechanical rules: communication. A symbolic formula such as Fs = kx provides both instructions for how to compute something— in this case, the force required to extend or compress a spring by a given length— and a way of conveying a proposition about the world.7
- 15. 4 ‹ Introduction It is my contention that the modern idea of algorithm, as the term is used in computer science, depends on a particular way of disentangling computa- tion from the complexities of communication that first took shape in the pure mathematics of the nineteenth century.8 Although machine learning systems are often called (confusingly) by the same name as the precisely defined procedures dealt with in the theory of algorithms, I hope to show that machine learning represents a break from this technical concept that places centuries- old epistemological boundaries in jeopardy. The history of algorithms has been told in both long and short ver- sions. In a broad sense, algorithmic thinking goes back at least as long as the written record.9 On clay tablets, the ancient Babylonians wrote down rule- based procedures for numerical computation in which the computer scientist Donald E. Knuth perceived the rudiments of his discipline.10 The word algorithm (early on spelled a range of ways, such as algorism, algorithmus, algram, or augrym) is less ancient but still very old— it was formed in the twelfth century from the name of the Arabic mathemati- cian Muḥammad ibn Mūsā al- Khwārizmī, who described techniques for computing with Hindi– Arabic numerals in the ninth century.11 These techniques— including the familiar addition, subtraction, multiplication, and division procedures one still learns in school— made up the original “algorithm.” As early as the sixteenth century, the word algorithm came to encompass a range of other techniques beyond these original ones, often involving symbolic algebra. In searching for precursors to the totalizing ambitions that now attend computation, popular histories commonly sin- gle out the German polymath Gottfried Wilhelm Leibniz. Starting in the 1660s, Leibniz attempted to create what he called a calculus ratiocinator—a system of symbolic “calculation” that could resolve disputes about virtu- ally any topic. The science writer Martin Davis describes the modern com- puter as a fulfillment of “Leibniz’s Dream” of extending mathematical sym- bol manipulation into a universal method that can be applied to anything whatsoever.12 More focused scholarship by historians including Michael Mahoney and Lorraine Daston has shown that such sweeping narratives overlook the ways computational practices have changed over the centuries.13 Mark Priestley has argued persuasively that computer programming has no in- trinsic relation to other fields such as symbolic logic but rather came to relate to them through intentional choices made by computer scientists.14 Matthew L. Jones and Maria Rosa Antognazza have placed Leibniz into historical context and showed that his work was not exactly algorithmic in the modern sense.15 This more historicist perspective has led to a contrast- ing narrative in which the concept of algorithm is very new. Venerable as
- 16. Introduction › 5 the word algorithm may be, its meaning arguably did not reach its modern form until the 1960s, when computer science emerged as an academic dis- cipline. The six authors of the book How Reason Almost Lost Its Mind have argued that algorithms were not a model of rationality until the Cold War period, when think tank researchers sought to replace human judgment with strictly rule- based decision- making.16 The algorithm’s rise to the sta- tus of a social concern is even more recent, stemming from a confluence of technical developments in machine learning with entrenched structures of inequality and discrimination.17 This historicization of the idea of algorithm should serve as a warn- ing against uncritically identifying the symbolic methods of the past with modern algorithms. The algorithm as we know it is a complex amalgam whose prehistory encompasses a range of practices, including astronomi- cal and statistical computation, bureaucratic procedures, market econom- ics, and governmental data- gathering efforts such as the US census. As a background to modern algorithms, symbolic methods are important less on account of their intrinsic relevance than because of the role they came to play in technical discourse. In the 1960s and ’70s, the discipline of com- puter science came to view algorithms as abstract processes that maintain a stable identity even as they are implemented, explained, applied, and in- terpreted in a range of ways. As I show in this book, this way of thinking is implicated in a long series of debates about the relation of symbols to language. Should the same symbols be used both to compute results and to present them to others? To what extent can their meanings be chosen at will, and to what extent does the establishment of meaning require social agreement? If a symbol is defined using words, does that entail that it in- herits the imprecision of natural language? Such issues would now be seen as extrinsic to computation, involving the significance people assign to algorithms, not the algorithms them- selves. But this boundary has not always been in place, as one can see by examining how what counted as an algorithm has changed over time. The Indian computational techniques have always involved instructions, taught either through direct imperative statements or by example, for what to do with symbols: if the sum is greater than 9, write a 1 above the digit to the left. As people recognized long before the computer age, this type of procedure can potentially be performed by machines.18 Yet the original “algorithm” also involved another, less obviously mechanical sort of rule: 9 means nine. The practice, that is, included rules not just for how to ma- nipulate the symbols but also for how to interpret them. While mathemati- cians long recognized that these semantic rules differed from calculating procedures, they were a part of the “algorithm” just the same.
- 17. 6 ‹ Introduction Symbolic algebra complicated these matters by introducing letters to indicate unspecified values, as in ax + b. This use of letters, introduced by François Viète in the 1590s, laid the groundwork for the modern algorithm by enabling procedures to be described in an abstract form that leaves the inputs unspecified. But these letters were linked together with operators such as + and –that were, at least early on, supposed to have fixed mean- ings. Establishing these meanings may not have posed a major problem in simple cases, but things became trickier as symbolic methods extended into theoretically fraught fields such as the infinitesimal calculus, and they became yet worse in utopian schemes like Leibniz’s attempt to develop symbolic methods for politics. Suppose, for instance, we introduce a sym- bol to denote equity. How can we be sure that everyone using this symbol agrees about what equity is? The importance given to conceptual clarity made it difficult to ignore the question of what it takes to make a symbol mean something, and disparate answers to this question had strong impli- cations for what symbolism could do. The expulsion of meaning from algorithms did not so much resolve these issues as divest them of epistemological significance. An early phase of this process may be discerned in the nineteenth century, when algebra- ists like George Boole granted formal rules a newly foundational role in their science. The boundary solidified in the twentieth century with the development of programming languages. Early programming languages such as ALGOL, first introduced in 1958, provided at once a way to control computers and a standard medium for publishing algorithms. As means of communication, programming languages do not, in general, work au- tonomously from the languages people speak; code typically uses words, both in built- in keywords like if and for and the user- defined names of functions and variables, to make its workings easier to understand. The received explanation of these linguistic inclusions is that they are mere conveniences that aid comprehension without affecting the algorithm it- self, which is defined in terms of a formal semantics. This division between “hard” algorithmic logic and “soft” communicational matters— a division that came to pervade the discourse of computer science— gives program- mers license to push ahead in the design of computational systems without worrying about what it would take to establish an accord about meaning, if, indeed, this accord is ever established at all. Historicizing the relation of symbolic methods to language shows that this way of thinking is not inherent to symbolic methods; things have been otherwise in the past, and they could be otherwise in the future. Language- based AI systems like GPT- 3, with their admixture of computational logic and collectively produced linguistic data, push the distinction between
- 18. Introduction › 7 computation and communication to its utmost limits, and they thus pro- vide an occasion to reconsider fundamental assumptions about how com- putational processes relate to language. The central claim of this book is that the modern idea of algorithm depends on a particular sort of subject– object divide: the separation of disciplinary standards of rigor from the complex array of cultural, linguistic, and pedagogical factors that go into making systems comprehensible to people. In the discipline of computer programming, these standards provide a way of thinking about computa- tional procedures— of creating them and judging them— that grants these procedures an objective existence as mathematical abstractions, apart from concrete computer systems. This subject– object divide is deeply em- bedded not just in textbook definitions of algorithm but also in the design of modern programming languages, which generally make algorithmic logic as independent as possible from matters of communication; this ab- straction facilitates the transfer of algorithms across computer systems and across application domains. This way of thinking was not firmly in place until the nineteenth century, and revisiting the conditions that produced it can help us better understand the implications of language- based machine learning systems like GPT- 3. The idea of algorithm is a levee holding back the social complexity of language, and it is about to break. This book is about the flood that inspired its construction. From Formulae to Source Code In broaching linguistic issues in relation to mathematics, this book joins a long tradition in the historiography of science. In the 1990s, scholars such as Peter Dear and Robert Markley drew attention to the role of language in the emergence of experimental science in the seventeenth century.19 More recent scholarship has explored the influence of linguistic disciplines from the past on mathematics and computation. There has, in particular, been a great deal of research on the intersection of linguistics with early computer history, including the importance of theories of syntax for programming languages and the emergence of machine translation as a research pro- gram.20 Looking at earlier time periods, scholars such as Kevin Lambert and Travis D. Williams have discussed mathematicians’ engagements with philology, which long concerned itself with the histories of mathematical symbols, and rhetoric, whose techniques can be discerned in mathemati- cal proofs.21 With some exceptions, histories of mathematical symbolism have fo- cused primarily on epistemological matters such as changing standards of mathematical proof, the new modes of thought opened by notations
- 19. 8 ‹ Introduction like ab , and how mathematical constructs relate (or do not relate) to real- ity. This book considers these matters, but it places more emphasis on the relatively neglected communicational side of symbolism. Communication, as the form of the word suggests, requires a common ground between peo- ple, and it is not self-evident that this common ground works the same way with words and symbols. For centuries, it has been recognized that the use of words is to some extent constrained by convention. As the seventeenth- century philosopher Bernard Lamy put it, “We might, if we please, call a Horse a Dog, and a Dog a Horse; but the Idea of the first being fixt already to the word Horse, and the latter to the word Dog, we cannot transpose them, nor take the one for the other, without an entire confusion to the Conversation of Mankind.”22 To communicate effectively in English, one must, at least broadly, follow the usages of others. The meanings of alge- braic symbols, on the other hand, appear to bend to the individual will: one can write, “let a = 5,” and that is what a will mean.23 To many observ- ers, such individualistically defined symbols have seemed, paradoxically, to convey ideas with a level of transparency that words could not match. Historical thinkers have addressed this apparent paradox in a range of ways, reflecting changing precepts about language, knowledge, and the formation of thought. An attention to these issues complicates received thinking about the role of algebraic symbolism in the origin of modern science. It has long been a common narrative that the Scientific Revolution of the seventeenth century involved the “mathematization” of the physical sciences. The trend more recently has been toward recognizing that the category of mathemat- ics itself changed in the period. The 2016 edited collection The Language of Nature: Reassessing the Mathematization of Natural Philosophy in the Seven- teenth Century works toward a more nuanced view of what it means for a science to be mathematized.24 The present study contributes to this nuanc- ing by examining the changing ways people made sense of mathematical symbols from the early modern period to present. While algebraic notation inspired a great deal of excitement in the seventeenth century, this excite- ment was not, as I hope to show, always tied to a conception of “mathemat- ics” at all. Early on, the excitement had more to do with the visual nature of the symbols, which promised a mode of communication fundamentally different from spoken languages such as English and Latin. Understanding the place of symbolic methods in the history of science thus requires histo- ricizing not only the category of mathematics but also the category of lan- guage; in particular, we must consider changing opinions on the relation of writing to speech and on how the common ground of communication should be established.
- 20. Introduction › 9 Attempting to find an absolute beginning for this history would be hopeless. Practices that look to us like algorithms have existed at least as long as writing itself, developing independently in a range of cultures. Aside from some background about ancient and medieval mathematics, my account starts in the sixteenth century, when the modern form of al- gebraic notation began to be codified. As I discuss in chapter 1, between the mid- sixteenth century and the mid- seventeenth, this notation revolu- tionized the practice of algebra: whereas equation- solving procedures had previously been expounded largely through words, one could now express them in compact formulae. Amid a general climate of suspicion toward language, such symbols came to be seen as a superior alternative, a way of presenting ideas directly to the eye without the mediation of words. This confidence in the transparency of symbols rested, I argue, on a belief that certain universal ideas were divinely etched onto all human minds, thus enabling perfect communication independently of the contingencies of language. Leibniz’s work was both a culmination of this early modern obsession with symbols and an inflection point. In chapter 2, I discuss the role of symbols in both his mathematical work and his attempt to create a calculus ratiocinator. Leibniz was one of the earlier writers (although not the first) to extend the word algorithm to something other than variants of the In- dian calculating techniques: he used its Latin and French cognates to refer to the differentiation procedure of his version of calculus. Yet his meaning was not quite the modern one, and an attention to this semantic nuance reveals an aspect of the history of algorithmic thinking that has often been overlooked. Leibniz modeled his “algorithm” not on common arithmetic but on symbolic algebra; it consists not of a precisely defined procedure that determines the correct manner of proceeding at each step but rather of a collection of equations for use in transforming expressions. This alge- braic sense of algorithm, which had widespread and enduring influence, placed the idea in an intimate relation to the development of new symbolic notations. Leibniz experimented with such notations in a wide range of contexts, from the ∫dx notation for integrals to binary numerals to attempts to de- velop symbolic methods for politics and law. Shifting ideas about lan- guage were, however, undermining the grounds of this project. In Han- nah Dawson’s account, a pivotal figure in linguistic thought was Leibniz’s intellectual rival John Locke.25 Leibniz’s dispute with Locke is commonly interpreted as epistemological, dealing with the legitimacy of nonempiri- cal forms of knowledge. The debate also, as I show in the chapter, had im- plications for symbolic methods. Leibniz assumed that concepts already
- 21. 10 ‹ Introduction existed in the mind at birth, so that stabilizing the meanings of symbols would not be a major problem. Locke troubled this assumption and, by doing so, called into question whether the symbols really were so different from words. In Locke’s long shadow, mathematicians paid a heightened attention to conceptual definitions; clarity, it was now believed, stemmed not from the notation itself but from the way mathematical concepts were formed in the mind. Although the rise of Lockean views of language spelled doom for Leib- niz’s more extreme claims about symbols, it disrupted neither the devel- opment of symbolic methods nor the desire to turn algebra into a univer- sal language. In chapter 3, I focus on a relatively little- discussed successor to Leibniz’s universal characteristic developed in the 1790s by Nicolas de Condorcet. At the height of the Reign of Terror following the French Revolution, Condorcet sketched out a system that would provide algebra- like notations for all manner of subjects. Like Leibniz, Condorcet was out to resolve people’s political and cultural differences by means of symbols. Yet his method was very different. Unlike Leibniz, Condorcet did not pre- sume that the ideas expressed by symbols were already universal; rather, he wanted to make them universal through a program of education. This approach rendered his system overtly politicized, dependent on a particu- lar vision of what society should look like. Although Condorcet’s scheme can be assigned little direct influence, it typifies a contention over the politics of symbolism that deserves a larger place in the historiography of computation. Standard accounts of eighteenth- century mathematics emphasize a division between national traditions: Continental mathematicians embraced Leibniz’s notation and method, whereas the English followed Isaac Newton in rejecting them. I argue in chapter 3 that eighteenth- century mathematics was cut across by ideological as well as national divides. As Sophia Rosenfeld has shown, language became a divisive topic during the French Revolution, as people blamed the Revolution’s splintering on a failure to agree on the meanings of such terms as liberty, equality, and fraternity.26 A central thinker in the linguistic thought of the period, the Abbé de Condillac, held up algebra as a model of the clarity needed to resolve such disagreements. Viewing algebra as, in Condillac’s terms, a “well- formed language” led to a number of debates over whether the symbols really did have clear definitions, as in a notorious controversy over the existence of negative numbers. How sym- bolic methods worked hinged, in this moment, on an issue regarding the politics of language: whether the meanings of signs ought to be governed collectively by the people or decided on by the learned. This conflict was less resolved than it was abandoned. In the early nine-
- 22. Introduction › 11 teenth century, algebraists turned their attention from conceptual defi- nitions to formal rules, which provided a new standard of mathematical rigor. In chapter 4, I focus on the work of George Boole, the English Irish mathematician who described the system that would eventually become Boolean logic. Boole’s work has seldom been considered part of the uni- versal language tradition exemplified by Leibniz and Condorcet, being typically positioned at the intersection of algebra and logic. But in the 1847 book in which he first introduced his system, Boole describes symbolic logic as “a step toward a philosophical language.”27 Taking this claim seri- ously, I contend that Boole’s project was enabled by another major shift in linguistic thought. While Boole was just an enamored with symbols as his precursors, he lacked their hostility toward words; instead, he espoused a respect and even a reverence toward the languages people inherit from their ancestors. This attitude enabled the two factions that clashed in the eighteenth century to arrive at a truce: instead of replacing language, the symbols were supposed to work together with it, at once drawing rigor from mechanical rules and meaning from words. The old antagonism toward language would soon enough return in the work of Gottlob Frege, Ernst Schröder, and Rudolf Carnap, who once again envisioned replacing words with symbols. But even their work did not undo the epistemological divisions that formed in Boole’s time. In chapter 5, I consider the early programming language ALGOL, whose name means “algorithmic language”—a choice that heralds the widespread adoption, starting around 1960, of the word algorithm as a general term for precisely defined computational procedures. ALGOL’s creators described it as a “universal language” that could specify algorithms in a form both readable by humans and executable by machines.28 But as with Boolean logic, ALGOL’s claims to universality are narrow. Rather than replacing the vernacular all the way down to the formation of actual human thought, ALGOL employs words (often in English) to help people understand pro- grams. What was supposed to be universal in ALGOL was only the algo- rithmic “essence” of a program, which was distinguished sharply from issues in which ordinary language still had to play a role, such as communi- cation and education—in short, from the aspects of computation that were coming to be known as “human factors.” The example of ALGOL shows that the algorithm as we now know it depends on a particular way of drawing disciplinary lines. When computer scientists started giving theoretical heft to the term algorithm, they were trying to identify essential elements of computational systems that could be analyzed mathematically, in isolation from the messiness of how those machines worked in their social contexts. This division between “hard”
- 23. 12 ‹ Introduction algorithmic matters and “soft” social ones remains deeply ingrained in the technical design of programming languages and the discourse surround- ing them. But it is not inevitable. Before the late nineteenth century, “algo- rithms” were not usually understood to exclude issues of communication; through Boole’s time, computational procedures typically included rules not just for what to do with symbols but also for what the symbols meant. How to establish this meaning was a matter for philosophical contention, and disparate views about language entailed divergent visions for what universal computation would be. It is primarily these earlier ways of thinking— the ones that are notice- ably different from modern computation— that I emphasize in this book. In the history of science, it is a methodological precept to avoid falling into the style called “Whig history”— to avoid, that is, describing histori- cal developments through linear narratives of progress that implicitly side with the positions that won. Histories of mathematical symbols tend to be extremely Whiggish, complimenting authors who use notations that later became standard and chastising those who do not. I certainly do not mean to deny the advantages of symbolism, but my purpose is less to celebrate it than to understand it, and I accordingly hope to describe what was lost with the adoption of symbols as well as what was gained. I also hope to show that the symbolic method is not a fixed category. The ways people have understood symbols changed multiple times over the centuries, and the modern idea of algorithm is a product of particular circumstances and epistemological commitments. Signs of another such change began to appear in the early twenty- first century. Over the course of the 2010s, the word algorithm came increas- ingly to refer not to the precisely defined procedures ALGOL was designed to represent but to machine learning systems like GPT-3. While the idea of machine learning has existed since the early computer era, this shift in the meaning of algorithm, as I argue in the coda, represents more of a break from twentieth-century conceptions than has generally been recognized.29 Text generators like GPT- 3 promise a new programming paradigm in which, instead of designing a computational procedure, programmers give the computer orders in English. Even for those (perhaps a minority) who are fully comfortable with this idea, it is hard to deny that its widespread adoption would give a renewed importance to the flaws of language— to the possibility that words are not actually clear or stable enough to form an adequate medium for technical knowledge. With the widespread adoption of machine learning, the division between “hard” logic and “soft” commu- nicational matters has become troubled, and algorithms have become a site of contestation.
- 24. Introduction › 13 New as these developments are, they in some ways mark a return to the situation in the eighteenth century, before Boole and his contempo- raries threw up a barrier between symbols and language. Mathematicians in the eighteenth century did not view the meanings of words as irrelevant to symbolic methods; instead, they heartily debated whether symbols had to correspond to received definitions of words or whether they could be defined anew. Nor did they set computational systems apart from politics. Some viewed symbols as a way of challenging received ways of thinking, an idea that came to be associated with the rationalizing reforms of the French Revolution. Others took the opposite view, cherishing words as a precious inheritance whose influence was needed to keep mathematical knowledge in line with the culture of a country. Attending to these earlier discourses, as this book aims to do, can provide us with a better sense of the possibilities and problems that exist at the intersection of computation and language. It may be helpful to think of this history as a succession of guiding terms— ideas that, in particular historical contexts, set the standards by which symbolic methods were judged. In the seventeenth century, Eu- ropeans typically described computation as an artifice or art, meaning a systematically developed set of skills. What made computation an art was its transmissibility: one could physically demonstrate, articulate, or write down the correct way of doing it, thus enabling people to develop and prac- tice the skill in a controlled fashion. In the eighteenth century, the valuing of artifice largely gave way to the cult of natural reason—a guiding principle that valued the mind’s inborn faculties. This way of thinking encouraged a deemphasis of explicit rules in favor of conceptual explanations that were supposed to make the correct way of performing a computation intuitively obvious. In Boole’s time, the reaction against Enlightenment thought led to a turn away from natural reason to the quite contrary valuing of culture. Under this star, the mechanical had to be balanced with the organic, and thus abstract mathematical systems and human thought, as fostered by the languages that develop in communities, formed two halves of a whole. While the idea of culture continues to influence computation, the idea guiding the modern algorithm is, if anything, technology. Technology is a very old word, but it once meant something very different from its pres- ent sense, referring either to a treatise about a skilled practice or to the set of technical terms used in discussing it.30 The modern meaning, which became dominant in the late nineteenth century, has more to do with the practical application of scientific knowledge. Viewing computation as technology encourages defining problems precisely so as to isolate aspects of systems that can be subjected to rigorous engineering methods— a per-
- 25. 14 ‹ Introduction spective that motivated early computer scientists to theorize algorithms as abstract procedures that may be analyzed apart from the specific contexts in which they are used. The full ramifications of this divide- and- conquer strategy did not become apparent until the early twenty- first century, when techniques that were developed within an intellectual framework that abstracted out almost all human experience became a force that runs much of the world. The history of symbolic methods is in some ways remote from the po- litical contentions that now surround algorithms. This book largely deals with a time when the idea of universal computation was more a mat- ter of starry- eyed speculation than a social reality. But many of the issues that arose from this speculation have remained with us in the computer age. Questions like whether symbolic methods can or should be politi- cally neutral have come up again and again over the centuries at moments when these methods were venturing into new territory. The terms of de- bate, however, have varied widely, and attending to earlier moments can be revealing about the assumptions of the present discourse. I begin in the early modern period, when excitement about symbolic methods was widespread— but for reasons quite opposed to those that have inspired the hype surrounding twenty- first- century AI.
- 26. 15 [ Chapter One ] Symbols and Language in the Early Modern Period The alphabet is really now superfluous for in this sign all men can find salvation. —Goethe, Faust, Part II (trans. Atkins) Idols and Hieroglyphs In the scientific circles of the seventeenth century, words had a bad reputa- tion. In the 1623 version of his book The Advancement of Learning, Francis Bacon warned against what he called the “idols of the market”— the “vul- gar” notions that, in everyday speech, tend to “insinuate themselves into the understanding” by means of words.1 As a protection against “the seduc- ing incantation of names,” he tentatively suggests definitions and “terms of art,” but even these are not enough; truly preventing words from “doing violence to the understanding,” he states, will require “a new and deeper remedy.”2 At almost exactly the same time, there was an explosion of new mathematical symbols.3 In the mid- 1500s, algebra often took the form of words, with even equations, which we now think of as made out of sym- bols, appearing in knotty prose. By the mid-1600s, this logorrhea had given way to compact symbolic expressions like ax + b = c. Although Bacon himself had little interest in mathematics, scholars have long noted an al- liance between these new symbols and his followers’ hostility toward lan- guage.4 Algebraic notation, brought into something like its modern form by Thomas Harriot and René Descartes in the early decades of the 1600s, came to be associated with a philosophical ideal of clarity, and numerous thinkers, G.W. Leibniz among them, envisioned developing analogous symbols for all manner of subjects. This chapter gives an overview of the symbolic methods that existed be-
- 27. 16 ‹ Chapter One fore Leibniz’s arrival on the scene in the 1660s. It focuses on two practices that would eventually form major sources for the modern idea of algo- rithm. The first is the set of techniques to which the word algorithm origi- nally referred. This word (then more commonly spelled algorism) gener- ally referred to the procedures of numerical computation that probably originated on the Indian subcontinent in the medieval period.5 The second is the algebraic symbolism that solidified in the early 1600s. Whereas it is now a cliché to call mathematical notation a universal language, early modern textbooks presented numerals and algebraic symbols less as lan- guage than as forms of writing comparable to the alphabet.6 Alphabetical writing, as the linguist Amalia E. Gnanadesikan explains, “is a transforma- tion of language, a technology applied to language, not language itself.”7 To their early modern advocates, symbols promised a way of improving the technology of writing so as to free it from the uncertainty of words. This view raised theoretical problems that would ultimately explode in the debate between Leibniz and John Locke, and that would render symbolic methods philosophically contentious for centuries. The early reception of symbolic algebra reflected a clash between con- ceptions of mathematical knowledge. As numerous scholars have shown, the question of what constituted “mathematics” was far from settled at the time; the category traditionally encompassed not just geometry and arith- metic but also astronomy and music, and some writers extended it to other practices such as the construction of machines.8 For many thinkers in the period, the heart of mathematics was Euclidean geometry. For instance, when Galileo Galilei made his famous statement— in his 1623 book The Assayer— that God wrote the book of the world in the language of math- ematics, he was explicitly referring to geometric diagrams, not to any sort of symbolic notation.9 Throughout the sixteenth and seventeenth centu- ries, Europeans held algebra in lower esteem than geometry, since it was not one of the traditional liberal arts and was perceived to lack rigorous standards of proof.10 Symbolic algebra transformed a range of practices in the seventeenth century, but its methods were widely regarded as practical rather than truly scientific, and they would long be hounded by conceptual difficulties. Going back to G.H.F. Nesselmann’s work in the nineteenth century, historians of mathematics have explained the development of algebraic symbolism with a three- stage model.11 First is the rhetorical phase, in which equations are presented entirely in words: “Three unknowns plus five equals twenty.” Next is the syncopated phase, in which some symbols are used as ligatures or abbreviations of words: “3 co. p. 5 eq. 20.” Finally, in the symbolic phase, the symbols replace words altogether and take on an
- 28. Symbols and Language in the Early Modern Period › 17 epistemological role: “3x + 5 = 20.” This model captures the gradualness of the process by which words gave way to symbols. Some of the basic alge- braic symbols originated as abbreviations: in his 1557 book The Whetstone of Witte, Robert Recorde explains the = sign as a way “to auoid the tediouse repetition of these woordes: is equalle to.”12 Such symbols, according to Nesselmann, eventually took on uses beyond merely shortening texts, making it possible to solve complex problems by transforming arrange- ments of symbols on a page. This three- stage account places much of early modern algebra in a gray area. As Albrecht Heeffer has argued, Nesselmann’s chronology is un- clear; the syncopated phase includes both ancient mathematicians such as Diophantus and early modern ones such as François Viète, ignoring the variety of mathematical practices that existed between them.13 Nes- selmann’s account also muddles the issues of what symbols people used— special signs such as + and = versus words such as plus and equals—and how they used the symbols. Viète employed a notation that mixed symbols with Latin words, which he even inflected in accordance with the rules of grammar— instead of =, he used æquatur. Yet he subjected these semiver- bal equations to rule- based transformations much like the ones now em- ployed in symbolic algebra. If we are looking for the origins of the style now known as symbol manipulation, then the transitions Nesselmann de- scribes are not necessarily pivotal. As far as problem- solving methods go, it makes little difference whether one manipulates words, abbreviations, or symbols. This revision of Nesselmann’s account, however, leaves an explanatory gap. Even if trading words for symbols had little effect on the procedures of algebra— on what one would now call the algorithms— symbolic nota- tion was not viewed as a minor development in the seventeenth century. Leibniz was far from the only one to see symbols as a basis for a univer- sal method; numerous thinkers, including Descartes and Isaac Newton, considered the possibility of doing for other fields what numerals and al- gebraic symbols had done for mathematics. Symbolic notation has an ob- vious advantage in its compactness, but this fact alone cannot explain the degree of the fervor. Advocates represented symbols as a way of putting thoughts directly on the page without mediation; algebraic notation was widely viewed as a way of circumventing Bacon’s idols of the market and granting knowledge a degree of certainty that words could not match. To understand these attitudes, we must contextualize the development of the notation not only in terms of the mathematical thought of the time, but also in terms of early modern ideas about language and writing. To do so, one must step into the mindset of a population for whom
- 29. 18 ‹ Chapter One reading and writing were not nearly as pervasive as they are today. Prior to the late seventeenth century, the word language primarily referred to spoken communication, and writing still seemed to many people, as Jona- than Hope puts it, “a strange technology,” a sometimes unreliable means of recording spoken words so that they could be recited later.14 The sixteenth century witnessed a number of attempts to make the technology of writ- ing more efficient and dependable. A 1588 book by Timothie Bright de- scribes an art of “characterie” that provides a means of “shorte, swifte, and secrete writing by Character.”15 Bright’s system consists of a large number of “Characters,” each of which has a “value, or signification” defined by a word (figure 1.1).16 By this means, one could write a whole word using no more space than a single letter. The shorthand movement was, as James Dougal Fleming has noted, entirely confined to England in the early mod- ern period, but shorthand- like practices existed in a range of languages.17 Alchemy and astrology, for instance, employed complex systems of sym- bols that were viewed as secret forms of writing akin to cryptographs and hieroglyphs.18 The fascination with these “characters” stemmed in part from the fact that they placed writing in a different relation to spoken language com- pared to alphabetical writing. In the early modern period, Europeans tended to discuss reading as if it involved a voice, be it literal or imagined.19 This way of thinking had a classical warrant, albeit one that was increas- ingly viewed as unsatisfactory. In On Interpretation, Aristotle describes the signification of written language as a multistage process: letters signify (spoken) words, words signify concepts, and (at least in the interpretation of some medieval readers) concepts signify things.20 Whereas the last step was endlessly controversial among the Scholastics, the first step was often glossed over. In early modern linguistic thought, it was common to use the word letter (in Latin, littera or litera) to refer indifferently to both alpha- betical characters and the speech sounds they represent.21 Nonphonetic symbols would seem to change this situation: Recorde’s “=,” for instance, does not in any obvious way represent the sounds of the words is equalle to. The Aristotelian model provides no clear guidance as to such symbols—the equals sign could be taken to signify the words in the manner of Bright’s “Characterie,” or else it could be seen as bypassing words and cutting straight to the concept of equality. In the case of numerals and algebraic symbols, there was a major mark in favor of the latter. Unlike alphabetical writing, these symbols could be read aloud in multiple languages: English speakers read “9 –1” as nine mi- nus one, whereas French speakers read it as neuf moins un, and the meaning
- 30. Symbols and Language in the Early Modern Period › 19 Figure 1.1. An example of an early modern shorthand notation, from Timothie Bright’s 1588 book Characterie. The Bodleian Libraries, University of Oxford, Douce W 3 (Weston Stack), sig. ¶3v. Images produced by ProQuest as part of Early English Books Online. www .proquest .com. Images published with permission of ProQuest. Further reproduction is prohibited without permission. appears to be the same in both cases. Borrowing from the Scholastic termi- nology, early modern thinkers explained such translinguistic symbols by distinguishing between nominal characters, which represented the sounds of words, and real characters, which referred directly to ideas or things. The idea of a real character appears most famously in Bacon’s 1623 Advance- ment of Learning, where it had particular reference to kanji— a subset of the Chinese han characters that can be read in either the Japanese or the
- 31. 20 ‹ Chapter One Chinese language.22 Bacon explains this quality as a departure from the Aristotelian model: the characters “express, not their letters [i.e., speech sounds] or words, but things and notions; insomuch, that numerous na- tions, though of quite different languages, yet, agreeing in the use of these characters, hold correspondence by writing.”23 As a result, “a book written in such characters may be read and interpreted by each nation in its own respective language.”24 Although Bacon does not discuss mathematical symbols in the passage, it later became routine to cite numerals and algebraic symbols as examples of real characters.25 For instance, the mathematician John Wallis wrote that, like Chinese characters, algebraic signs “so little need the interven- tion of Words to make known their meaning, that, when different persons come to express, in Words, the sense of those Characters, they will as lit- tle agree upon the same Words, though all express the same sense, as two Translators of one and the same Book into another Language.”26 Robert Hooke made a similar comparison with regard to “Arithmetical Figures.”27 Since it does not align with how modern linguistics understands writing, a number of modern scholars have dismissed the idea of a real character as mistaken or absurd; Jaap Maat goes so far as to call the idea a “myth.”28 But the idea is not wholly senseless when applied to mathematical sym- bols. Calling these symbols real characters amounts to claiming that they express universal ideas that are accessible to all people, regardless of what words one chooses when pronouncing them— that the English word mi- nus and the French word moins really share a common core of meaning by means of which the minus sign can be used to communicate the idea of subtraction clearly across languages.29 This is not to say that the real- character idea still holds water. The idea depended on a faith that the human mind was divinely constructed to mir- ror the world, which precludes any serious recognition of cultural diver- sity. Not everyone thought this way in the seventeenth century. But nu- merals and algebraic symbols really are, in a sense, more like an alternative to the alphabet than a language. As the next section shows, seventeenth- century textbooks taught numerals in a way that emphasized physical pen skills and speaking numbers aloud. This pedagogy had more in common with learning to read and write (or with learning a shorthand like Bright’s “characterie”) than with learning a second language. There was also a dif- ference in regard to gender: vernacular tongues were largely learned from women— from mothers and nurses— whereas writing and mathematics alike were both typically learned from male teachers.30 Examining how numerals were taught sheds light on why early modern philosophers such as Leibniz put so much stock in the power of symbols— and on how the
- 32. Symbols and Language in the Early Modern Period › 21 algorism of their time was different from algorithmic thinking as we now know it. The Meaning of Algorism That computation has anything to do with writing is not a given. Pre modern cultures developed a wide array of calculating implements, from abaci to the intricate, multilevel counting tables developed by the Inca. The rise of the Hindi– Arabic numeral system, however, gave computation a strong link to writing that would long implicate it in philosophical debates about language. This system probably originated on the Indian subconti- nent; the astronomer Brahmagupta described it in Sanskrit verse around 628 CE, although there is evidence that it was already in use prior to his work.31 The numerals later spread to the Arabic world, where they were described by the mathematician and astronomer Muḥammad ibn Mūsā al- Khwārizmī.32 While al- Khwārizmī did not invent these methods, it was his name that inspired the word algorithm, and so it is worth considering the contents of his work. Probably born in what is now Uzbekistan, al- Khwārizmī secured a posi- tion at the House of Wisdom, a library in Baghdad, where he wrote on a number of topics.33 His c. 820 Compendious Book on Calculation by Comple- tion and Balancing is largely about solving equations; the word algebra is derived from the word al-jabr (usually translated as “completion”) in the Arabic title of this book.34 His work on arithmetic, unfortunately, survives only in unreliable Latin translations.35 The most famous of these transla- tions is sometimes called “Dixit Algorizmi” (“Algorizmi said”) because the translator inserted that phrase at the beginnings of the first two para- graphs; this bit of scribal happenstance is thought to be the origin of the Latin word algorithmus and thus, ultimately, of the English word algorithm. Fromwhatwecangatherfromthesurvivingtranslations,al-Khwārizmī’s arithmetic book presented procedures for addition, subtraction, multipli- cation, division, halving, and doubling.36 In the “Dixit Algorizmi” version, the explanation of addition begins like this: You will add each place to the place that is above it with regard to its own kind, i.e., units to units and tens to tens. When ten has been collected in one of the places, i.e., in the place of the units or tens or in some other place, put a one instead of it and elevate it to a higher place, i.e., if you have ten in the first place which is the place of the units, make a one of it and raise it to the place of the tens and there it will signify ten. But if there remains something from the number that is less than X or the num-
- 33. 22 ‹ Chapter One ber itself is less than X, leave it in the same place. And if nothing remains, put a circle (i.e., o), so that the place may not be empty.37 This procedure contains many of the hallmarks of the intellectual style that eventually came to bear al- Khwārizmī’s name, including the use of conditional, if– then logic and even what we might think of as a loop: “Do likewise,” we are instructed, “also in all the places.”38 This is an archetypal algorithm: early computer scientists took it as a model for the sort of pro- cedure that a machine can be programmed to perform. Through the seventeenth century, the word algorism or algorithm still referred primarily to variants of this particular set of procedures. (While the spelling with a th appeared in Latin as early as the 1480s and in English in the 1650s, I will refer to these historical practices as algorism so as to avoid confusion.39) Some authors distinguished algorism from arithmetic, which was one of the seven liberal arts set out by the medieval philosopher Boethius. Arithmetic, in the Boethian sense, was about types of numbers: squares, primes, perfect numbers, and a range of others that are less well remembered.40 As the Elizabethan polymath John Dee put it, the purpose of this study was “arise, clime, ascend, and mount vp (with Speculatiue winges) in spirit, to behold in the Glas of Creation, the Forme of Formes, the Exemplar Number of all thinges Numerable: both visible and inuisible, mortall and immortall, Corporeall and Spirituall.”41 In contrast to such lofty doctrines, algorism was seen as one of the lower branches of math- ematics. While it was sometimes taught in Latin schools, the art of com- putation was primarily the business of the (mostly) men E.G.R. Taylor dubbed “mathematical practitioners”— people who taught mathematics independently of the university system through textbooks, lecturing, and tutoring.42 The methods they taught had applications in navigation, trade, and finance, in artisanal trades such as bricklaying and construction, and in military practices such as ballistics. In these practical fields, the Hindi– Arabic algorism competed with and sometimes worked together with a range of other forms of computation. The practitioners sold instruments such as the sector, which consisted of a hinged pair of rulers inscribed with scales that could be used to perform approximate calculations. Abaci and counting stones had a long history, and some people continued to prefer them; counting stones were espe- cially popular in Germany, and abaci would remain in use for centuries in eastern Europe. Calculation could also involve numerical tables, in which one could look up certain values without having to compute them oneself. The early seventeenth century saw a major advance in such techniques with the development of logarithm tables, which were introduced in 1614
- 34. Symbols and Language in the Early Modern Period › 23 by John Napier.43 Since adding the logarithms of two numbers produces the logarithm of their product, one could use logarithm tables to reduce multiplication to the much easier operation of addition. This technique was the basis of some of the period’s most advanced mathematical instru- ments, such as William Oughtred’s “circles of proportion,” a precursor of the slide rule that he described in 1632.44 Apart from teaching and selling instruments, the practitioners also pub- lished books from which one could, at least in principle, learn the art of calculation. Most of these books started with “common algorism,” mean- ing the use of ordinary counting numbers. This doctrine began with “nu- meration” or “notation,” which meant learning the meanings of the digits; afterward came discussions of operations such as addition, subtraction, multiplication, and division.45 The exact list varied. Some texts followed al- Khwārizmī in including special procedures for halving and doubling, and some added further operations such as extracting roots. Some textbooks also included special “algorisms” for calculations involving currency as well as more advanced ones for rational numbers and decimal fractions. Algorism also included additional procedures intended to verify results, since (no doubt) human computers would often make mistakes. The procedures described in these books resemble modern algorithms in their use of rigidly defined steps that begin and end with arrangements of symbols. It is this rigidity that gives the procedures the mechanical qual- ity that inspired Blaise Pascal and Leibniz to build calculating machines. But the first part of algorism— numeration— is different. According to Jo- hann Lantz’s 1616 arithmetic text, which Leibniz encountered in school, numeration is “the enunciation and expression of whatever number is set forth.”46 (The Latin enunciatio can mean either pronunciation or proposi- tion, suggesting a conflation of words and ideas similar to that of the Greek logos.) The first step was to familiarize oneself with the nine digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 as well as the “cipher” 0, which was viewed as a mere place- holder that had no inherent meaning. Students had to be able to recognize and inscribe these symbols dependably; algorism thus, as Jessica Otis has argued, required the basic pen skills that were a part of literacy.47 They also had to learn the symbols’ values, which were often taught through tables similar to the one by which Bright defined the “significations” of his short- hand characters (figure 1.2). They also had to learn the rules by which nu- merals are composed into numbers so that they could translate them into the number words of a language, as 84 becomes “eighty-four.” An extended discussion of this translation appears in another text Leib- niz studied in detail: Johann Heinrich Alsted’s 1630 Encyclopaedia.48 Alsted was educated at the Herborn Academy, which was a center of pansophism,
- 35. Figure 1.2. Hindi– Arabic numerals explained by means of roman ones. From Nicolaus Kauffunger’s 1647 German- language textbook Plenaria Arithmetica, p. 8. Kauffunger explains that numeration (Numeriren) teaches students how they “actually should correctly and tidily write and pronounce each number, just as, in grammar, orthography teaches correct writing” (2; translation mine).
- 36. Symbols and Language in the Early Modern Period › 25 an educational movement that emphasized making knowledge accessible to all; his encyclopedia provided a model for Leibniz’s own encyclopedic endeavors. Leibniz judged Alsted’s treatment of mathematics to be merely “average for his time,” but it is useful as an example of what would have been considered typical in the mid- seventeenth century.49 In his chapter on arithmetic, Alsted states that the digits are like an “Arithmetical alpha- bet.”50 Like Lantz, he associates understanding this alphabet with translat- ing the symbols into words: numeration, he writes, is “the right enuncia- tion of rightly written numbers.”51 He explains several techniques for this translation, including ways of marking the symbols up so as to make the translation easier: to help make sense of 89765878910, for instance, one can draw dots above or below every third digit after the first, going right to left, as in 89̇765̇878̇910. While Alsted is concerned with the “right enunciation” of numbers, his point is not that there is only one correct way to do it. The Greeks and Ro- mans, he observes, expressed numbers in various ways that are often much more verbose than the numeration of modern Latin. According to Alsted, Pliny the Elder might have expressed the number of soldiers in Xerxes’s army, 5,283,220, as (to translate loosely) “fifty times and twice a hundred and eighty- three thousands, two hundred twenty,” whereas one would now write “five double thousands, two hundred eighty- three thousands, two hundred twenty.”52 Likewise, markup procedures can produce differ- ent readings depending on how they are done: 10 ̣ 000 ̣ 000 ̣ 000 leads to “ten thousand thousand thousand,” whereas 100 ̣ 00 ̣ 000 ̣ 000 leads to “a hundred hundred thousand thousand.”53 Grouping the symbols by threes is, Alsted tells us, the easiest way of doing it, but he makes it clear that, whichever words one chooses, the number itself remains the same. This mediation between symbols and words has been overlooked in ac- counts of how language and mathematics related in the early modern pe- riod. Walter Ong and Robert Markley each have argued that early mod- ern mathematics was antidialogic— that it suppressed the back- and- forth interchange that was valued in classical thought.54 In Phaedrus, Plato has Socrates argue that speech is superior to writing because, in conversation, one can dynamically respond to questioning.55 When one reads an expla- nation off a sheet of paper, the words are always the same, but when one explains something one genuinely understands, the words come out differ- ently every time. Measuring intelligence by the ability to hold a conversa- tion (an idea that persisted all the way to Alan Turing) implies that merely having a written text at hand is no guarantee that one knows anything. One might suppose that numerals, in their alienation from the phonetics
- 37. 26 ‹ Chapter One of spoken languages, are even more inimical to this Socratic conception of knowledge than alphabetical writing. But in the pedagogical scene evoked by algorism texts, the opposite is the case. Contrary to Ong’s association of mathematics with a “silent object world,” numerals were not supposed to be contemplated mutely.56 Instead, they enable the teacher to prod the student to speak: sure, it is that many thousands, but how many hundreds? How many tens? Yet this proliferation of verbalizations had its limits. It may be equally ac- ceptable to read 4206 as “four thousand two hundred and six” or as “forty- two oh- six,” but reading it as “twelve” would simply be wrong. Teaching people to understand digits required establishing an accord about their values, which provided the common ground on which the dialogue took place. Alsted does not directly reference Bacon’s idea of a real character in this passage, but his account of the “signification” of numerals reveals a similar way of thinking. Just like kanji characters, numerals enable peo- ple from many nations to “read and interpret,” as Bacon put it, a text in their own languages; yet the numbers to which the symbols refer remain the same (we are supposed to assume) regardless of what language or what specific wording one chooses. The numerals thus offload, as it were, the signifying function that is ordinarily handled by languages like Latin onto the basic operating system of written communication: the alphabet itself. This background makes it easier to understand Leibniz’s confidence that the power of “calculation” could be extended to other areas. While the calculus ratiocinator is often discussed together with Leibniz’s work on calculating machines, the two are distinct, and it is not clear that the “cal- culation” by which he hoped to settle disputes was supposed to encompass only those aspects of calculation that a machine could perform. Knowing how to numerate correctly— being able to choose the right digits at the be- ginning of the process and read off the results correctly at the end— was also part of what calculation meant in his time, and using a machine did not render this knowledge irrelevant. Crucially, the fact that numeration was less mechanical than the rules of operation did not imply that it was any less certain. Alsted’s account of numeration suggests that the Hindi– Arabic system can express numbers in such a way that there can be no doubt about their true values. It thus provides reason to think that symbols could extend a similar level of certainty to other areas. This confidence in the power of symbols depends, however, on the stability of the ideas or things those symbols are supposed to signify. One might pronounce the symbol 8 with either the English eight or the Latin octo, but are the meanings of these two words really the same? If we place the languages into their historical contexts, they are arguably not: the Ro-
- 38. Symbols and Language in the Early Modern Period › 27 mans did not understand numbers the same way we do today. This may seem a pedantic distinction, but the problem is more glaring in domains that are more obviously contentious than common arithmetic. If we devise a character, for instance, to signify grace, it is far from a given that everyone will understand its meaning in the same way. As the next section shows, such issues were not specific to utopian schemes like Leibniz’s. They also arose in another major part of al- Khwārizmī’s mathematical work: algebra. While it had clear practical use, algebra also presented answers that could not be “said” in the way the results of algorism could. Its adoption thus un- settled the peaceful relationship numerals instated between symbols and language, raising theoretical difficulties that would recur in discussions of symbolic methods for centuries. Unspeakable Numbers The fact that the words algebra and algorithm derive from the work of the same person should not be taken to mean they have any necessary con- nection. If algebra means a way of solving equations— which is what the word almost always meant before the nineteenth century— then its history goes back millennia. Geometric equation- solving methods existed in Mes- opotamia and ancient Greece; Chinese mathematicians began developing numerical techniques for equation solving around 200 BCE.57 Diophantus of Alexandria anticipated some elements of symbolic algebra in the third century CE.58 That such practices are algorithmic is far from a given, and modern algebra, indeed, contains much that is not. But algebra did play an important role in the development of algorithmic thinking. Al- Khwārizmī opens his account of algebra by placing it under the scope of calculation: “When I considered what people generally want in calculating, I found that it always is a number.”59 Algebra, in this form, is an art of calculation that produces numbers— not one that is identical to algorism but one that can work together with it. Al- Khwārizmī’s version of algebra survives in the work titled The Com- pendious Book on Calculation by Completion and Balancing. His book be- gins with definitions and general techniques for equation solving along with geometric proofs, which he performed in the style of Mesopotamian geometry.60 The two basic techniques are al-jabr (الجرب), which means completion or restoration, and al-muqābala (َةلَبَاقُمْلا), which is translated in this context as balancing. While the word algebra derives from al-jabr, it now encompasses both of these techniques— moving terms from one side of an equation to the other (al-jabr) and canceling terms (al-muqābala). Employing these two techniques, al- Khwārizmī describes procedures for
- 39. 28 ‹ Chapter One solving six types of equation; he then discusses mercantile calculations, techniques for computing the areas of geometric figures, and calculations involving inheritance. This final section is indicative of the uses to which algebra was put in al- Khwārizmī’s context. The Abbasid Caliphate had complex rules for inheritance that required fulfilling certain equation- like conditions; al- Khwārizmī’s discussion of these issues takes up almost half the book.61 Like the surviving translations of al- Khwārizmī’s arithmetic, the alge- bra is oriented toward teaching the reader a set of practical methods. In contrast to the direct instructions of the algorism, he explains his alge- braic methods through specific problems, such as “half of a square and five roots are equal to twenty- eight dirhems [a unit of currency].”62 That is, in modern notation, x2 /2 + 5x = 28. His solution for this problem begins as follows: “Your first business must be to complete your square, so that it amounts to one whole square. This you effect by doubling it. Therefore double it, and double also that which is added to it, as well as what is equal to it. Then you have a square and ten roots, equal to fifty- six dirhems.”63 What he is instructing us to do here— doubling the coefficients— is specific to this instance of the problem: if the first number is something other than a half, one has to multiply by some other number. Yet this example is meant to stand in for a broader class of problems. “Proceed in this manner,” he concludes the section, “whenever you meet with squares and roots that are equal to simple numbers: for it will always answer.”64 Al-Khwārizmī’s compiling of these instructions marks the beginning of what Victor J. Katz and Karen Hunger Parshall call the “algorithmic stage” in the history of algebra— a stage in which algebra consisted primarily of procedures for solving particular classes of problem.65 Like algorism, algebra was long regarded as a practical matter in Eu- rope. Al- Khwārizmī himself would have had no strong reason to classify his work as either practical or theoretical— Islam, as it was interpreted by the Abbasids, encouraged a continuum between secular and holy knowl- edge.66 European universities had comparatively rigid hierarchies of pres- tige, and algebra did not self- evidently deserve the status of a learned dis- cipline. Unlike arithmetic and geometry, algebra had no place among the seven liberal arts; its diffusion in Europe was largely due to merchants such as Leonardo of Pisa (Fibonacci), who encountered Arabic mathematics while accompanying his father on a trading expedition to North Africa and described algebraic methods in his 1202 Book of Calculation.67 During the Renaissance, the Italian mercantile academies known as “abacus schools” taught algebra alongside a range of calculating practices.68 While algebra was sometimes taught at universities, its academic status long remained
- 40. Symbols and Language in the Early Modern Period › 29 less secure than those of geometry and arithmetic.69 Even in Leibniz’s life- time, powerful figures such as Isaac Barrow, who became the first Lucasian professor of mathematics at Cambridge at 1663, were dismissing algebra as a mere problem- solving technique lacking the force of demonstration.70 Algebra’s initial lack of prestige resulted not just from its association with commerce but also from conflicting ideas of number. In the early modern period, European mathematics was heavily under the sway of Euclid, along with other classical Greek thinkers such as Archimedes and Plato.71 Whereas we now tend to think of cardinal number as a singular concept, Euclid employs two number- like concepts that he treats as en- tirely distinct: quantity and magnitude. A quantity, for Euclid, is a num- ber of things, such as four apples, whereas a magnitude is a length, area, or volume. Euclidean magnitudes can also be compared by means of ratios, which he treats as distinct from the magnitudes themselves; this distinc- tion persists in the use of the different notations a/b = c/d and a : b :: c : d. Euclid also differentiated magnitudes by dimension, which algebra vio- lated in its use of square numbers: the expression x2 + x would seem to add an area to a length, which, in Euclidean terms, is nonsense.72 Arabic mathematicians had extensive access to Greek sources, so their work should not be seen as a wholly separate tradition. But the al-jabr and al-muqābala did in some ways clash with Euclidean ideas of number. When solving a quadratic equation, one has to take square roots. The situ- ation was clear enough when the root was rational: √ — 16 = 4. Yet one could also end up with a root whose exact value cannot be pinned down. In book 10 of the Elements, Euclid proves the existence of lines whose lengths can- not be expressed as multiples of any common unit.73 The proportion of a square’s diagonal to one of its sides, for instance, is √ — 2, whose value is not quite equal to any fraction. Such numbers arose frequently in solving qua- dratic equations and posed a problem for attempts to “calculate” an exact numerical solution. Euclid provided a suggestive term for these numbers: alogos (άλογος), or, as it might be translated, unspeakable.74 If the ratio between two mag- nitudes is irrational, this term suggests, then its true value cannot be said. This suggestion of a lack of speech was widely known in the medieval and early modern periods. While a number of thinkers, including the Persian astronomer Jamshīd al- Kāshī, had developed methods for calculating roots, this could never be done exactly in such cases.75 A marginal note in a copy of al- Khwārizmī’s algebra book states that one must be content with “an approximation, and not the exact truth: for God alone knows what the exact root is.”76 Such attitudes were common among Islamic mathemati- cians as well as some Christians such as Nicolas of Cusa and Jacques Pele-
- 41. 30 ‹ Chapter One tier, who both connected mathematics to a notion of divine mystery.77 Others, however, were more disturbed by the apparent unspeakability of irrational quantities, whose use in algebra did not seem to measure up to Euclidean or Archimedean standards of knowledge. One potential resolution was to forget about speaking the numerical value and be satisfied with saying “the square root of two.” A suggestive, albeit equivocal, example appears in Michael Stifel’s 1544 book Arith- metica integra.78 “It is justly disputed of irrational numbers,” Stifel writes, “whether they are true numbers, or fictions.”79 In favor of the existence of irrationals is their utility in calculation, on account of which “we are moved and compelled to confess, that they truly exist, namely by their ef- fects, which we perceive to be real, certain, and constant.”80 On the other hand, when we “try to subject them to numeration, and proportion them by rational numbers, we find that they flee perpetually.”81 The idea that the numbers “flee” (a subtly violent metaphor) exemplifies a characteristically Protestant emphasis on clear apprehension as a standard of mathematical truth. A personal friend of Martin Luther who was repeatedly imprisoned for his bold expressions of support, Stifel was not content to take matters on authority; he wanted knowledge whose force one could perceive.82 In spite of these reservations, Stifel developed a method, which he calls an “algorithm” (Algorithmus), for manipulating irrationals using a nota- tion somewhat like the modern √ — 2.83 For instance, in modern notation, to compute √ — 18 + √ — 8, one first determines the ratio between the two roots, which in this case turns out to be rational: 18 8 9 4 3 2 / / / = = . The sum of the roots, then, must stand in proportion to √ — 8 as 3 + 2 is to 2; hence, the sum, as we would now write it, is equal to 8 2 2 2 (3+2) / . Based on this, Stifel determines that √ — 18 + √ — 8 = √ — 50.84 He also includes a more ad- vanced “algorithm” for working with composites of different types of num- ber, such as 6 + √ — 12.85 This work (which has roots in Euclid’s discussion of “unspeakable” numbers) indicates that the term algorithm was already, in the sixteenth century, beginning to expand beyond its original reference to the Hindi– Arabic methods of computation into a broader category of symbolic method. In this case, it is notable that the methods in question deal with exact relations of irrationals, not numerical approximations— a quality that aligns them more with algebra than with computation. As the example of Stifel indicates, European mathematicians were, by the sixteenth century, feeling the limitations of the classical Greek number theories that had long reigned in universities. One of the earliest explicit repudiations of these theories appeared a few decades later in the work of the Flemish mathematician Simon Stevin. In 1585, Stevin published a
- 42. Symbols and Language in the Early Modern Period › 31 pamphlet whose title is variously translated as The Tenth or The Tithe, in which he introduced decimal notation (versions of which were already known in the Middle East and in China) to Europe.86 Rather than a decimal point, he used circled numbers to indicate the significance of each digit: 3①7②5③9④ means three tenths, seven hundredths, five thousandths, and nine ten- thousandths, corresponding to what we would write today as 0.3759.87 He goes on to show that a method much like common algorism may be used to perform operations with these numbers; he provides ap- plications in surveying, measurement, and mercantilism. Although Stevin’s circled numbers may have been cumbersome com- pared to the modern decimal point, his work led to a major shift in con- ceptions of number. For Stevin, a number is constructed not by measuring lines or counting indivisible units, but through the digits themselves— the tools of the commoners who practiced algorism. As a result, Stevin main- tains that “there are no absurd, irrational, irregular, inexplicable, or surd numbers.”88 Perhaps one cannot write √ — 2 as a fraction, but one can write it (in modern notation) as 1.41421356 . . . and continue the expansion as far as one likes. In 1594, Stevin described a procedure that can do just that: pinning down the root of an equation digit by digit by progressively divid- ing the number line into tenths.89 (Centuries later, Turing would prove the existence of “uncomputable” numbers, as Gregory Chaitin put it, whose digits cannot be generated through any clearly defined procedure; Stevin was not on as steady ground as he thought.90) While there is some debate about the exact nature of Stevin’s numbers, his work points in the direction of what is now called the real number continuum, a number concept that breaks entirely with classical theories. Stevin’s redefinition of number does not, however, encompass every numerical entity algebra can produce. A procedure like Stevin’s can ap- proximate the roots of positive numbers, including compounds such as 6 6 – , but it cannot account for the roots of negatives, which can read- ily emerge from algebraic methods. If, for instance, one were to apply al- Khwārizmī’s equation-solving method to the equation we would now write as x2 + (10 –x)2 = 48, one gets 5 25 26 5 1 ± = ± – – .91 Unlike irrationals, the square root of negative one cannot even be approximated by decimal fractions, since no positive or negative number has a negative square. This result could simply be rejected as a sign that the equation has no solution, and for centuries this would remain the typical reaction. Yet the fact that the procedures of algebra could produce such results seemed to imply that the procedures themselves were ill founded, and the problems worsened as mathematicians attempted to extend their range.
- 43. 32 ‹ Chapter One The best- known instance of this issue appears in the work of the Italian polymath Gerolamo Cardano. In his 1545 book The Great Art, or, the Rules of Algebra, Cardano described a complete solution for cubic equations— that is, in modern notation, equations of the form ax3 + bx2 + cx + d = 0. The desire for such a solution had been long- standing; the Persian mathe- matician Omar Khayyam (best known as the presumed author of the poetic cycle the Rubaiyat) had analyzed cubic equations in the eleventh century, but neither he nor his immediate followers could find an algorithmic solu- tion.92 Cardano’s solution was attended by a well- known scandal that gives a taste of what life as a mathematician was like in sixteenth- century Italy. Cardano learned part of the solution, as he acknowledges, from an unpub- lished poem shared by an acquaintance known as Tartaglia (which means “the stammerer”). Tartaglia swore him to secrecy about this result, and yet Cardano published it anyway. Cardano had an excuse. Tartaglia, he had dis- covered, was not the first to discover the result; another mathematician named Scipione del Ferro had discovered it over a decade before. Cardano took this to mean it was fair game to publish. Yet he failed to acknowledge the oath, which led to a bitter dispute culminating in a Renaissance alterna- tive to a duel: a public mathematics contest between Tartaglia and one of Cardano’s students, Ludovico Ferrari, who bested the stammerer and put the matter to an end. The solution Cardano assembled breaks the problem down into more specific cases such as x3 + ax = b and x3 = ax2 + b. This division into cases is necessary because, like al- Khwārizmī, Cardano does not allow equa- tions to have negative coefficients; it is thus impossible to combine all cu- bic equations into one general form. His “rules” take the form of knotty prose with the occasional use of an abbreviation, such as ℞ for “root.” To solve x3 + ax = b, for instance, one follows these instructions: Cube one- third the coefficient of the number of things, add it to the square of one- half the constant of the equation; & take the square root of the whole. You will duplicate this, and to one of the two you add the one- half of a number you have already squared and from the other you subtract the same. You will then have a binomium and its apotome. Then, subtracting the cube ℞ of the apotome from the cube ℞ of the binomium, the remainder [or] that which is left is the value of the thing.93 The historian of mathematics Helena M. Pycior characterizes these pro- cedures as “prose algorithms,” and they can readily be interpreted as al- gorithms in the modern sense.94 Unlike al- Khwārizmī and Stifel, Cardano
- 44. Symbols and Language in the Early Modern Period › 33 presents the rules in general forms rather than by means of specific exam- ples, pre figuring the abstraction that would later come to characterize the programming language. But Cardano’s theory is not wholly algorithmic. Along with each “rule,” he presents a geometric demonstration of its va- lidity, which he includes “so that, beyond mere experimental knowledge, reasoning may reinforce belief” in the results.95 The goal of his book, then, was not just to compile procedures for solving practical problems; the reader was also supposed to come away with an understanding of why those procedures were right. Such, at least, was the ideal. The geometric basis of Cardano’s methods ran into trouble in the so- called irreducible case. As Pycior points out, cer- tain cubic equations, such as the innocent-looking y3 = 8y + 3, have rational solutions that one cannot find via Cardano’s methods without encounter- ing the square roots of negatives.96 In this case, applying Cardano’s rule gives (in modern notation) this rather cumbersome expression: y = + + 3 3 3 2 1805 108 3 2 1805 108 – – – Using a procedure like Stifel’s “algorithm of composite irrational numbers,” one can reduce this value to y = 3, which clearly does satisfy the original equation.97 This result is correct by the standards of twenty-first-century al- gebra, but the majority of mathematicians in the sixteenth century viewed such expressions as nonsense. Some of Cardano’s early followers, such as Raphael Bombelli, managed to overcome their reservations; square roots of negative numbers, Bombelli wrote, initially “seemed to me to be based more on sophism than on truth, but I searched until I found the proof.”98 Yet the theoretical basis for such proofs remained highly uncertain. Cardano’s tentative foray into new realms of number, it should be em- phasized, did not depend on the use of symbols. Cardano was undeniably practicing algebraic reasoning, but apart from numerals and a few abbre- viations, he explained his procedures in words. This aspect of algebra was soon to change. Already in the mid- 1500s, the convoluted prose of medi- eval equation- solving procedures was starting to give way to compact for- mulae. By the mid- 1600s, symbols had taken over. As I discuss in the next section, this new notation gained much of its power from the use of letters: x, y, z for unknown values and a, b, c for known ones. Unlike numerals, with their fixed meanings, these symbols bear different values with every problem solved. In his 1650 textbook Arithmetick, Jonas Moore explains the difference this way: numerals are “Notation certain, and determinate,” whereas algebraic symbols are “Notation uncertain, undeterminate, and ar-
- 45. 34 ‹ Chapter One bitrary.”99 These “uncertain” symbols seemed to many thinkers, paradoxi- cally, to achieve a level of clarity words could not match— a development that almost immediately inspired dreams of a universal method. From Numbers to Letters In today’s primary schools, the transition from arithmetic to algebra is marked by the sudden appearance of letters. After years of learning about fixed values such as 4 + 5, the student is suddenly confronted with expres- sions such as 4 + x and must learn how to reason about a value that is not yet known. The need for some way of referring to unknown quantities is essential to algebra, but the use of letters to fill this office is a relatively re- cent development.100 Modern algebraic notation is based on an idea that would later become central to the design of programming languages— the use of arbitrary symbols to represent values that are left unspecified for the purpose of generalizing a procedure. This use of symbols has become deeply entrenched in modern algorithmic thinking, but it was not there in the work of al- Khwārizmī himself, and it came at the cost of placing com- putational procedures in a fraught relation to meaning. Early iterations of Arabic algebra most commonly represented un- known quantities with words. Medieval Arabic writers referred to the un- known as shay (ءْ َ ش), meaning thing; in Latin, this became res, which is the word Cardano uses. But long before Cardano’s time, other ways of express- ing unknowns had appeared. Robert of Chester’s c. 1145 Latin translation of the Al-jabr includes a condensed summary of the “rules” (regul[a]e) of the art that uses symbols not in the Arabic original. In his critical edition, Barabas B. Hughes approximates these symbols as ø, ϑ, and ʒ. For instance: “When ø is equal to ϑ and ʒ, ø and ϑ must be divided by ʒ, ϑ halved, the half drawn into itself, the product added to the number. The radix of the aggre- gated whole minus half ϑ reveals what is sought.”101 The letters correspond to coefficients attached to terms of specific degrees: ø is a constant, ϑ the unknown, ʒ the square of the unknown. In modern terms, then, the equa- tion is ʒx2 + ϑx = ø. Yet placing Chester’s symbols in the company of modern notation like that is misleading. His ø, ϑ, and ʒ are not interchangeable tokens like the as and bs of modern algebra; they function more like common nouns in that they have, to borrow a pair of terms from Gottlob Frege, both sense and reference.102 In Frege’s terms, reference is what is singled out: the reference of the phrase the liar, for instance, is the person being called a liar. Sense is the semantic freight the words carry: in this case, all it is to be a liar. Ches- ter’s symbols employ both types of meaning, at once referring to numeri-
- 46. Symbols and Language in the Early Modern Period › 35 cal values of the coefficients and conveying information about how those values fit into the equation— about, that is, which value is applied to the root and which to the square of the root. There is, to be sure, something of sense in the symbols of modern algebra; René Descartes instated a conven- tion of using x, y, and z for unknowns and a, b, and c for knowns, thus using the letters as hints as to the purposes different values play. But the rules of symbol manipulation do not depend on these conventions; a and x obey identical rules. Chester’s “rule,” on the other hand, is incomprehensible without a recognition of the different senses of the symbols. The transition from Chester’s ø, ϑ, and ʒ to our as and xs was not a linear process. In his 1494 Summa de arithmetica, Luca Pacioli presented equa- tions in a compact form that abbreviated unknowns as “co.,” after the Ital- ian cosa (thing).103 This word inspired a type of symbol that came to be known as the “cossic character,” which was placed next to a number to in- dicate a certain power of the unknown. These symbols first appeared in Christoff Rudolff’s c. 1525 algebra textbook, the shortened title of which— Die Coss— led a generation of Germans to refer to algebra with a term that means, etymologically, “the thing.”104 Rudolff’s symbols were later adopted by Stifel and Recorde, while others employed similar notations with different symbols (figure 1.3). Whereas Chester’s symbols represent unspecified coefficients, cossic symbols accompany coefficients to indicate Figure 1.3. Robert Recorde’s explanation of cossic characters from The Whetstone of Witte (1557), sig. S.i.v. Call no. 56546, Rare Books, The Huntington Library, San Marino, California. The first character is a unit indicating a constant value; the second is a unit equal to the unknown, the third to the square of the unknown, and so forth. He continues the sequence further on the next page.
- 47. 36 ‹ Chapter One Figure 1.4. The first set of equations presented in Robert Recorde’s The Whetstone of Witte, sig. Ff.i.v. Call no. 56546, Rare Books, The Huntington Library, San Marino, California. the degree of the term. The symbols are thus, in a sense, less abstract than Chester’s ø, ϑ, and ʒ, providing a notation for the unknown but offering no way to leave the parameters of a problem unspecified so that the solution can be stated as a general rule. Combined with Recorde’s = sign, cossic characters enabled the con- struction of what are sometimes characterized as the first- ever equations (figure 1.4). On account of the absence of words in Recorde’s version of cossic notation, Heeffer argues that his algebra was fully symbolic.105 But Recorde’s symbols are, from a semantic perspective, quite different from the ones we are used to. In modern notation, the left- hand side of the first equation in the figure would be 14x + 15— an expression whose value is indeterminate until x is fixed. This indeterminacy is alien to Re- corde’s algebra.106 Instead, Recorde explains cossic characters as units of measurement— one talks about five roots or five squares in the same way one talks about “20. shippes.”107 As in common algorism, the cossic charac- ters have a “numeration” that involves translating them into words— in this case, we might say something like fourteen roots more fifteen nombers.108 These words express not an indeterminate quantity but rather a “com- pounde nomber” that has, as far as Recorde is concerned, a fixed value as much as twelve does.109 While he employed symbols, Recorde was still concerned primarily with representing numbers, which he understood as numbers of things. As far as algorithmic thinking goes, a crucial turning point occurred in the work of the French lawyer, councillor to Kings Henri III and IV, and amateur astronomer François Viète. In his 1591 Introduction to the Art of
- 48. Symbols and Language in the Early Modern Period › 37 Analysis, Viète introduced a notation that kept the units but did away with numbers: G A +B Z B G in planum in quadratum in This would be, in modern notation, ga bz bg + 2 .110 Yet the notations are not quite equivalent. The “planum” specifies that A is two- dimensional, which is necessary because, for Viète, lengths and areas are different types of value that may not be added together.111 If these dimensions indicate an attach- ment to classical ideas of number, Viète’s use of letters pointed in a new direction. In contrast to the cossic notations, Viète used letters to repre- sent both the unknowns and the knowns. He used vowels to represent un- knowns and consonants for knowns in order, as he wrote, “that this work may be assisted by some art.”112 These letters, which he called “species,” enabled a practice known in English as “specious arithmetic” (specious was probably pronounced with a hard c, as in Latin). Instead of working on single equations such as 14x + 15 = 71, one could now draw conclusions about general “species” of equation such as Ax + B = C. There have been a number of differing accounts of where Viète’s As and Bs came from, and it is not clear that they have any connection at all to the cossic characters used by Rudolff and Recorde.113 In his 1685 Treatise of Algebra, John Wallis linked the letters to Viète’s background in law, argu- ing that they originated from a way lawyers abbreviated people’s names.114 In his classic study of the history of number concepts, Jacob Klein treats them as wholly novel, arguing that Viète initiated the turn away from clas- sical conceptions of mathematics toward “symbolic formalism.”115 The best recent account of Viète is by Jeffrey A. Oaks, who argues that Klein over- looked the geometric basis of Viète’s method; in Oaks’s account, Viète was drawing on the practice of using letters to refer to elements of geometric diagrams.116 There is, however, a difference worth noting. Of necessity, geometric diagrams show a figure with particular proportions, even when the drawing is meant to stand in for a general class of figure. One can draw many scalene triangles with various proportions, but one cannot draw a scalene triangle with indeterminate proportions. Viète’s notation, on the other hand, enables procedures to be described abstractly, without the need for any particular example. Viète’s idea emerged as a part of the humanist push to recover suppos- edly lost forms of ancient knowledge. There had, in particular, been a re- vival of interest in the third- century work of Diophantus of Alexandria, along with the later Alexandrians Theon and Pappus. Diophantus’s work was preserved by Arabic scholars, but it was not widely read in Europe
- 49. 38 ‹ Chapter One until the 1570s, when portions of it were translated into Latin by Raphael Bombelli.117 Probably drawing on a combination of ancient Greek, Egyp- tian, and Babylonian mathematical traditions, Diophantus had developed methods for solving simple equations as well as indeterminate systems of equations— that is, systems that do not impose enough conditions to ex- clude the existence of infinitely many solutions.118 His solutions, which he presents in a compact notation somewhat like cossic characters, involve making arbitrary assumptions about the relations of values and then work- ing out their consequences.119 For instance, to solve the problem of find- ing two squares that sum to 16, he assumed that their roots stood in the relation, as we would write it now, y = 2x + 4.120 On this assumption, the original equation may be solved to produce a solution to the problem: 256 25 and 144 25 . Fairly or not (almost certainly not), Viète positioned himself as restoring the theory of equations to a pristine Alexandrian state. This art, he wrote, has been “spoiled and defiled by the barbarians,” on account of which he must get “rid of all its pseudo- technical terms (pseudo- categorematis) lest it should retain its filth and continue to stink in the old way.”121 The unsubtle subtext is that he wishes to expunge Arabic sources in favor of Hellenistic ones, a chauvinism manifest in his dislike of the term algebra.122 Viète hoped to replace al- Khwārizmī’s techniques with what he took to be Diophantus’s secret method, which he called analysis.123 In Viète’s defini- tion, which he attributes to Theon of Alexandria, analysis starts by assum- ing what is sought and working out its consequences, whereas synthesis starts from what is true and deduces other truths.124 Viète divided analysis into three phases. The first is the zetetic, which involves the use of specious arithmetic to derive a formula for the unknown in terms of what is known. The second phase, the poristic, involves the formulation and proof of the resulting solution. Last is the exegetic or rhetic phase, in which “from the equation set up or the proportion there is produced the magnitude itself which is sought.”125 Rhetic analysis, in other words, is the “let A = 5” mo- ment: the moment in which one sets the values of the letters and resolves the formula into a number. In his later work Zetetics (1593), Viète presents a large number of worked- out problems that give a better sense of how this method actually worked than the general explanations of the Introduction. For instance, he considers this problem: “Given the difference of lines, & difference of cubes: to find the lines.”126 That is, find two numbers that differ by B, and whose cubes differ by D. Viète solves it like so. Call the sum of the numbers E; then E + B is twice the larger number and E − B is twice the smaller number. Through some algebraic manipulation, he ends up with this:
- 50. Symbols and Language in the Early Modern Period › 39 D sol. , B cubo. quatur E quadrato 4 – B3 æ That is, as we would now write, 4D B E – 3 2 3B = . Next, he restates the formula in words: “The quadruple difference of cubes, minus the cube of the differ- ence of lines, if applied to triple the difference of lines: appears the square of the aggregate line.”127 Finally, he plugs in the values B = 6 and D solidum = 504 to get a numerical answer. In a fact often glossed over, he presents the result in an entirely different notation from the one he used to manipu- late equations: “summa laterum 1N, 1Q æquatur 100.”128 Much like cossic characters, this notation treats the unknown value as a unit of measure- ment. In this case, N is a unit set to equal the “sum of the lines” and Q is another unit equal to its square; thus, the square of the sum is 100. From this, we can deduce that the sum itself is 10, and two numbers that satisfy the problem are 2 and 8.129 The approach parallels that of Diophantus, but with a difference: whereas Diophantus uses numbers all along, Viète does most of the work with letters, only plugging in the numbers at the end. Viète was grandiose in his ambitions for this art. He ends the 1591 Intro- duction with an imperialistic statement of purpose set in caps: “To leave no problem unsolved.”130 This statement signals the arrival— right alongside the first glimmerings of modern algebraic symbolism— of uni- versalizing ambitions for what those symbols could do. There is a clear res- onance between Viète’s notation and programming languages and, indeed, a historical line of influence linking the two. But the road, again, is not lin- ear. As the example in the foregoing paragraph shows, what Viète was do- ing was not quite symbolic algebra as we know it. The verbal statement of the formula seems, from a modern point of view, redundant— why bother with words if you already have the symbols? The rhetic phase, in which he plugs in the numbers to get an answer, raises an issue that is subtler but es- pecially indicative of the problems that would come to face universal com- putation schemes. The verb let—in Latin, Viète uses the jussive subjunctive sit, as was common in geometric proofs— indicates a mediation between the undetermined Bs and Ds of specious arithmetic and the realm of num- bers. Viète’s self- consciousness about this mediation is apparent from the fact that the instant he “lets” the letters have values, he switches to a differ- ent notation more aligned with classical ideas of number. At the rhetic mo- ment, then, the symbols change in nature, ceasing to be abstract instruc- tions and coming to represent numbers of things. All this complexity would soon vanish. While Viète’s idea of analy- sis would have a lasting influence, the most consequential element of his work was the letters, which were soon extracted from their verbal set- tings. Among their earliest and most enthusiastic adopters was Thomas
- 51. 40 ‹ Chapter One Figure 1.5. Thomas Harriot’s explanation of the rules for division in his algebra system, from the 1631 Artis analiticae praxis. © British Library Board, General Reference Collection C.74.e.4., p. 10. Images produced by ProQuest as part of Early English Books Online. www .proquest .com. Images published with permission of ProQuest. Further reproduction is prohibited without permission. Harriot.131 Harriot was best known for accompanying Sir Walter Raleigh to Roanoke Island and for writing a book about his experiences in Virginia. His algebraic work was not published until 1631, about a decade after his death, although it probably circulated in manuscript form earlier.132 After a preface and a few pages of definitions in Latin, the 1631 volume con- sists almost entirely of symbols. Harriot developed a simplified version of Viète’s notation that expunged the Latin words: in place of A cubum, he wrote aaa (figure 1.5). The preface to the book, probably written by Wal- ter Warner, states that Viète had performed his analyses with “interpreted signs,” whereas Harriot found it more convenient to use “a literal notation: that is, the letters of the alphabet.”133 Harriot also broke down the linguistic barriers by which Viète had so carefully demarcated the three phases of analysis. No more would cossic notations reappear once numbers enter the
- 52. Symbols and Language in the Early Modern Period › 41 scene; now, the same symbols could be used for both algebraic manipula- tion and numerical computation. It did not take long for admirers of these new symbols to conceive of ex- tending them to other areas. In the 1630s, John Pell attempted to develop the ideas of Viète and Harriot into a more complete and rigorous theory of equations.134 Pell connected this interest in algebra to a fascination with novel forms of communication such as shorthand, in which his father- in- law, Henry Reynolds, was deeply interested.135 As did a number of others at the time, Pell thought specious arithmetic could provide a model for a new form of writing that could express anything in symbols.136 The idea was to divide concepts up into simple components— he thought fire, for instance, combined “hot thing” and “shining”— and develop symbols for those sim- ple components that could be placed together in various configurations.137 The idea of such an “art of combinations” was very old, but Viète’s use of the term analysis provided a new way of explaining it. Whether the analy- sis of concepts into parts really had anything to do with algebraic analy- sis would long be a topic of debate. Regardless, the example of Pell shows that Viète’s early followers were interested in more than solving numerical problems— they also saw something in his method that could be applied to any number of topics of inquiry. Harriot and Pell were both located in England, where the mania for symbols was strongest in the early seventeenth century. Across the English Channel, Pierre de Fermat, Marino Ghetaldi, and Jean- Louis Vaulezard applied Viète’s methods to geometric problems.138 In regard to symbols, however, the crucial Continental figure was René Descartes, whose 1637 book Geometry introduced a notation recognizably like the one we use to- day. It was Descartes who popularized the convention of using x, y, and z for unknowns as well as the exponent notation x3 . Apart from being more compact than Harriot’s aaa, this notation eventually opened the possibility of exponents other than integers and, in particular, of a way of unifying ex- ponents and roots: √ — x = x½ . Descartes claimed not to have been influenced by Viète at all, a statement that remains controversial among scholars.139 Whatever the case may be, he contributed further to the establishment of a fully “literal” notation for expressing formulae. One practical issue of the new notation was an important precursor to the modern algorithm: the formula. Viète’s “species” made it possible to explain at least some types of computational procedure entirely through symbols. As an example, take Michael Dary’s Interest Epitomized, Both Compound and Simple (1677). This text presents rules for computations regarding compound interest in a compact form (figure 1.6). The proce- dure shown in the figure, which gives the principal of a loan based on the
- 53. 42 ‹ Chapter One Figure 1.6. A computational procedure for working with compound interest, first presented as a formula and then worked out as an example. Note that the notation r(t) indicates what we would now write as rt . The calculation is done using base ten logarithms; the logarithm used for u is slightly inaccurate, but the error does not affect the rounded answer. From Michael Dary’s 1677 book Interest Epitomized. Cambridge University Library M.6.29, p. 2. Reproduced by kind permission of the Syndics of Cambridge University Library. amount owed after the accrual of compound interest, is expressed in an ab- stract formula—(a = u r t ( ) )— that one can apply in short order. The compila- tion of such computational rules aligned with the Baconian attitudes of ex- perimentalists like Robert Hooke, with whom Dary was friendly.140 Bacon thought that science would be incomprehensible to the masses but would nonetheless produce new techniques that would be useful to them.141 Sym- bolic formulae fit this model nicely: mathematical geniuses could develop theories that would trickle down to commoners in the form of practical operations that they could use without having to trouble their heads about how it all worked.142 Accounts of the emergence of algebraic notation have shown a strong tendency toward Whiggish narratives that presume the superiority of sym- bolic methods. If only Arabic algebraists had developed symbols, Katz and Parshall speculate, they could have surpassed their European counter- parts.143 But the superiority of symbolic methods is not self- evident. The Islamic mathematicians Viète denigrated as “barbarians” did gain access to Diophantus’s work— indeed, they had more of it than Europeans would see until the 1970s— and yet they paid little attention to the symbols.144
- 54. Symbols and Language in the Early Modern Period › 43 Perhaps they were right in that. Viète’s program of abstraction— solving problems by the hundred rather than one at a time— opened the way for the totalizing ambitions that now give the word algorithmic an increas- ingly ominous ring. That these ambitions should develop in the way they ultimately did, however, was not predestined. In pursuing the universal method Viète had promised, seventeenth- century mathematicians plot- ted out two divergent paths, neither of which quite aligns with algorithmic thinking as we know it. Out of the Covert of Words Did the new symbols make algebra easier to understand? It depends on what one means by understand. Advocates emphasized the compactness of symbolic notation, which, as Robert Hooke wrote, “is of huge Use in the Prosecution of Ratiocination and Inquiry, and is of vast Help to the Under- standing and Memory.”145 These cognitive advantages were not specific to mathematical fields: Hooke wishes that natural history could be expressed in a similar shorthand employing “as few Letters or Characters as it has con- siderable Circumstances.”146 William Oughtred, an influential mathemati- cal practitioner who popularized specious arithmetic in England, went fur- ther. In the 1647 book The Key of the Mathematicks New Forged and Filed (a revised English translation of his earlier Latin textbook), he suggests that symbols can reveal the naked truth beneath the veil of language: “Where- fore that I might more cleerly behold the things themselves, I uncasing the Propositions and Demonstrations out of their covert of words, designed them in notes and species appearing to the very eye.”147 By claiming to be “uncasing” ideas from words, Oughtred positions the symbols as a neutral medium devoid of rhetoric—the naked truth, unadorned, all substance, no style. If so, they would seem a perfect remedy to the Baconian complaint about language. Yet the symbols did not always appear clear or comprehensible. One of Oughtred’s pupils, the Oxford astronomer Seth Ward, notes the potential opacity of symbols in a 1654 book on trigonometry: “It does not escape me that this little book, which abstains for the most part from words and strives to carry the things themselves to the understanding at once, will be seen by some as portentous and difficult: Truly it was produced chiefly for the sake of those to whom (provided they did not neglect themselves) designations of this kind have already become familiar.”148 This passage exhibits a tension that would continue to pervade discussions of symbolic methods for centuries, all the way to the emergence of the programming language. The symbols seemed to represent mathematical ideas with preci-
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- 56. solemnly and constantly denied; and from this, as a foul reproach, he has been cleared by the moderns, and particularly by Martin Luther, who lays the whole blame of this controversy on the turbulent and angry Cyril. (See Hypostatic Union.) The discordancy not only between the Nestorians and other Christians, but also among themselves, arose, no doubt, in a great measure, from the ambiguity of the Greek terms hypostasis and prosopon. The councils assembled at Seleucia on this occasion decreed that in Christ there were two hypostases. But this word, unhappily, was used both for person and subsistence, or existence; hence the difficulty and ambiguity: and of these hypostases it is said the one was divine, and the other human;--the divine Word, and the man Jesus. Now of these two hypostases it is added, they had only one barsopa, the original term used by Nestorius, and usually translated by the Greeks, person;” but to avoid the appearance of an express contradiction, Dr. Mosheim translates this barbarous word aspect,” as meaning a union of will and affection, rather than of nature or of person. And thus the Nestorians are charged with rejecting the union of two natures in one person, from their peculiar manner of expressing themselves, though they absolutely denied the charge. In the earliest ages of Nestorianism, the various branches of that numerous and powerful sect were under the spiritual jurisdiction of the Catholic patriarch of Babylon,--a vague appellation which has been successively applied to the sees of Seleucia, Ctesiphon, and Bagdad,--but who now resides at Mousul. In the sixteenth century the Nestorians were divided into two sects; for in 1551 a warm dispute arose among them about the creation of a new patriarch, Simeon Barmamas, or Barmana, being proposed by one party, and Sulaka, otherwise named Siud, earnestly desired by the other; when the latter, to support his pretensions the more effectually, repaired to Rome, and was consecrated patriarch in 1553, by Pope Julius III., whose jurisdiction he had acknowledged, and to whose commands he had promised unlimited submission and obedience. Upon this new Chaldean patriarch’s return to his own country, Julius sent with him several persons skilled in the Syriac language, to assist him in establishing and extending the papal empire among the Nestorians;
- 57. and from that time, that unhappy people have been divided into two factions, and have often been involved in the greatest dangers and difficulties, by the jarring sentiments and perpetual quarrels of their patriarchs. In 1555, Simeon Denha, archbishop of Gelu, adopted the party of the fugitive patriarch, who had embraced the communion of the Latin church; and, being afterward chosen patriarch himself, he fixed his residence in the city of Van, or Ormia, in the mountainous parts of Persia, where his successors still continue, and are all distinguished by the name of Simeon; but they seem of late to have withdrawn themselves from their communion with the church of Rome. The great Nestorian pontiffs who form the opposite party, and who have, since 1559, been distinguished by the general denomination of Elias, and reside constantly at Mousul, look with a hostile eye on this little patriarch; but since 1617 the bishops of Ormus have been in so low and declining a state, both in opulence and credit, that they are no longer in a condition to excite the envy of their brethren at Mousul, whose spiritual dominion is very extensive, taking in great part of Asia, and comprehending within its circuit the Arabian Nestorians, as also the Christians of St. Thomas, who dwell along the coast of Malabar. NETHINIMS. The Nethinims were servants who had been given up to the service of the tabernacle and temple, to perform the meanest and most laborious services therein, in supplying wood and water. At first the Gibeonites were appointed to this service, Joshua ix, 27. Afterward the Canaanites who surrendered themselves, and whose lives were spared, were consigned to the performance of the same duties. We read, Ezra viii, 20, that the Nethinims were slaves devoted by David and the other princes to the ministry of the temple; and elsewhere, that they were slaves given by Solomon; the children of Solomon’s servants, Ezra ii, 58; and we see, in 1 Kings ix, 20, 21, that this prince had subdued the remains of the Canaanites, and had constrained them to several servitudes; and, it is very probable, he gave a good number of them to the priests and Levites for the service of the temple. The Nethinims were carried into captivity with the tribe of Judah, and there were great numbers of them near the coast of the Caspian Sea, from whence Ezra brought
- 58. some of them back, Ezra viii, 17. After the return from the captivity, they dwelt in the cities appointed them, Ezra ii, 17. There were some of them also at Jerusalem, who inhabited that part of the city called Ophel, Neh. iii, 26. Those who returned with Ezra were to the number of two hundred and twenty, Ezra viii, 20; and those who followed Zerubbabel made up three hundred and ninety-two, Ezra ii, 58. This number was but small in regard to the offices that were imposed on them; so that we find them afterward instituting a solemnity called Xylophoria, in which the people carried wood to the temple with great ceremony, to keep up the fire on the altar of burnt sacrifices. NETTLES. We find this name given to two different words in the original. The first is חרול, Job xxx, 7; Proverbs xxiv, 31; Zeph. ii, 9. It is not easy to determine what species of plant is here meant. From the passage in Job, the nettle could not be intended; for a plant is referred to large enough for people to take shelter under. The following extract from Denon’s Travels may help to illustrate the text, and show to what an uncomfortable retreat those vagabonds must have resorted. One of the inconveniences of the vegetable thickets of Egypt is, that it is difficult to remain in them; as nine- tenths of the trees and the plants are armed with inexorable thorns, which suffer only an unquiet enjoyment of the shadow which is so constantly desirable, from the precaution necessary to guard against them.” The קימוש, Prov. xxiv, 31; Isaiah xxxiv, 13; Hosea ix, 6; is by the Vulgate rendered urtica,” which is well defended by Celsius, and very probably means the nettle.” NICE or NICENE CREED is so denominated, because the greater part of it, namely, as far as the words, Holy Ghost,” was drawn up and agreed to at the council of Nice, or Nicæa, in Bithynia, A. D. 325. This council was assembled against Arius, who, though he brought down the Son to the condition of a creature, inferior, for that reason, in nature to the Father, yet acknowledged his personal subsistence before the world, and his superiority in nature to all the things that were created by him. So that there was need of some higher expression in this case than the other, to import his equal dignity of nature with the Father and Creator of all; and nothing was
- 59. found to answer the purpose so well as the term ὁμοούσιος. The rest of this creed was added at the council of Constantinople, A. D. 581, except the words, and the Son,” which follow the words, who proceedeth from the Father,” and they were inserted A. D. 447. The addition made at Constantinople was caused by the denial of the divinity of the Holy Ghost by Macedonius and his followers; and the creed, thus enlarged, was immediately received by all orthodox Christians. The insertion of the words, and the Son,” was made by the Spanish bishops; and they were soon after adopted by the Christians in France. The bishops of Rome for some time refused to admit these words into the creed; but at last, A. D. 883, when Nicholas the First was pope, they were allowed, and from that time they have stood in the Nicene creed, in all the western churches; but the Greek church has never received them. See Arius. NICODEMUS, a disciple of Jesus Christ, a Jew by nation, and a Pharisee, John iii, 1, &c. At the time when the priests and Pharisees had sent officers to seize Jesus, Nicodemus declared himself openly in his favour, John vii, 45, &c; and still more so when he went with Joseph of Arimathea to pay the last duties to his body, which they took down from the cross, embalmed, and laid in a sepulchre. NICOLAITANS. St. John says in his Revelation, to the angel of the church of Ephesus, But this thou hast, that thou hatest the deeds of the Nicolaitans, which I also hate,” Rev. ii, 6; and again, to the angel of the church of Pergamos: So hast thou also them that hold the doctrine of the Nicolaitans, which thing I hate,” Rev. ii, 15. These are the only two places where the Nicolaitans are mentioned in the New Testament: and it might appear at first, that little could be inferred from these concerning either their doctrine or their practice. It is asserted, however, by all the fathers, that the Nicolaitans were a branch of the Gnostics: and the epistles, which were addressed by St. John to the seven Asiatic churches, may perhaps lead us to the same conclusion. Thus to the church at Ephesus he writes: Thou hast tried them which say they are Apostles and are not, and hast found them liars,” Rev. ii, 2. This may be understood of the Gnostic teachers, who falsely called themselves Christians, and who would be not unlikely to assume also the title of Apostles. It appears from
- 60. this and other passages, that they had distinguished themselves at Ephesus; and it is when writing to that church, that St. John mentions the Nicolaitans. Again, when writing to the church at Smyrna, he says: I know the blasphemy of them which say they are Jews, and are not, but are the synagogue of Satan,” Rev. ii, 9. The Gnostics borrowed many doctrines from the Jews, and thought by this means to attract both the Jews and Christians. We might therefore infer, even without the testimony of the fathers, that the Gnostic doctrines were prevalent in these churches, where St. John speaks of the Nicolaitans: and if so, we have a still more specific indication of their doctrine and practice, when we find St. John saying to the church in Pergamos, I have a few things against thee, because thou hast there them that hold the doctrine of Balaam, who taught Balak to cast a stumbling block before the children of Israel, to eat things sacrificed unto idols, and to commit fornication,” Rev. ii, 14. Then follow the words already quoted, So hast thou also them that hold the doctrine of the Nicolaitans, which thing I hate.” There seems here to be some comparison between the doctrine of Balaam and that of the Nicolaitans: and I would also point out, that to the church in Thyatira the Apostle writes, I have a few things against thee, because thou sufferest that woman Jezebel, which calleth herself a prophetess, to teach and to seduce my servants to commit fornication, and to eat things sacrificed unto idols,” Rev. ii, 20. The two passages are very similar, and may enable us to throw some light upon the history of the Nicolaitans. Tertullian has preserved a tradition, that the person here spoken of as Jezebel was a female heretic, who taught what she had learned from the Nicolaitans: and whether the tradition be true or not, it seems certain, that to eat things sacrificed unto idols, and to commit fornication, was part of the practice of the Nicolaitans. These two sins are compared to the doctrine of Balaam: and though the Bible tells us little of Balaam’s history, beyond his prophecies and his death, yet we can collect enough to enable us to explain this allusion of St. John. We read, that when Israel abode in Shittim, the people began to commit whoredom with the daughters of Moab: and they,” that is, the women, called the people unto the
- 61. sacrifices of their gods: and the people did eat, and bowed down to their gods,” Num. xxv, 1, 2. But we read farther, that when the Midianites were spoiled and Balaam slain, Moses said of the women who were taken, Behold, these caused the children of Israel, through the counsel of Balaam, to commit trespass against the Lord in the matter of Peor,” Num. xxxi, 16. This, then, was the insidious policy and advice of Balaam. When he found that he was prohibited by God from cursing Israel, he advised Balak to seduce the Israelites by the women of Moab, and thus to entice them to the sacrifices of their gods. This is what St. John calls the doctrine of Balaam,” or the wicked artifice which he taught the king of Moab: and so he says, that in the church of Pergamos there were some who held the doctrine of the Nicolaitans. We have therefore the testimony of St. John, as well as of the fathers, that the lives of the Nicolaitans were profligate and vicious; to which we may add, that they ate things sacrificed to idols. This is expressly said of Basilides and Valentinus, two celebrated leaders of Gnostic sects: and we perhaps are not going too far, if we infer from St. John, that the Nicolaitans were the first who enticed the Christians to this impious practice, and obtained from thence the distinction of their peculiar celebrity. Their motive for such conduct is very evident. They wished to gain proselytes to their doctrines; and they therefore taught that it was lawful to indulge the passions, and that there was no harm in partaking of an idol sacrifice. This had now become the test to which Christians must submit, if they wished to escape persecution: and the Nicolaitans sought to gain converts by telling them that they might still believe in Jesus though they ate of things sacrificed unto idols.” The fear of death would shake the faith of some; others would be gained over by sensual arguments: and thus many unhappy Christians of the Asiatic churches were found by St. John in the ranks of the Nicolaitans. We might wish perhaps to know at what time the sect of the Nicolaitans began; but we cannot define it accurately. If Irenæus is correct in saying that it preceded by a considerable time the heresy of Cerinthus, and that the Cerinthian heresy was a principal cause of St. John writing his Gospel, it follows, that the Nicolaitans were in
- 62. existence at least some years before the time of their being mentioned in the Revelation; and the persecution under Domitian, which was the cause of St. John being sent to Patmos, may have been the time which enabled the Nicolaitans to exhibit their principles. Irenæus indeed adds, that St. John directed his Gospel against the Nicolaitans as well as against Cerinthus: and the comparison which is made between their doctrine and that of Balaam, may perhaps authorize us to refer to this sect what is said in the second Epistle of St. Peter. The whole passage contains marked allusions to Gnostic teachers. There is another question concerning the Nicolaitans, which has excited much discussion. It is a question entirely of evidence and detail; and the two points to be considered are, 1. Whether the Nicolaitans derived their name from Nicolas of Antioch, who was one of the seven deacons: 2. Supposing this to be the fact, whether Nicolas had disgraced himself by sensual indulgence. Those writers who have endeavoured to clear the character of Nicolas have generally tried also to prove that he was not the man whom the Nicolaitans claimed as their head. But the one point may be true without the other: and the evidence is so overwhelming, which states that Nicolas the deacon was at least the person intended by the Nicolaitans, that it is difficult to come to any other conclusion upon the subject. We must not deny that some of the fathers have also charged him with falling into vicious habits, and thus affording too true a support to the heretics who claimed him as their leader. These writers, however, are of a late date; and some, who are much more ancient, have entirely acquitted him, and furnished an explanation of the calumnies which attach to his name. We know that the Gnostics were not ashamed to claim as their founders the Apostles, or friends of the Apostles. The same may have been the case with Nicolas the deacon; and though we allow, that if the Nicolaitans were distinguished as a sect some time before the end of the century, the probability is lessened that his name was thus abused; yet if his career was a short one, his history, like that of the other deacons, would soon be forgotten: and the same fertile invention, which gave rise in the two first centuries to so many
- 63. apocryphal Gospels, may also have led the Nicolaitans to give a false character to him whose name they had assumed. NICOPOLIS, a city of Epirus, on the gulf of Ambracia, whither, as some think, St. Paul wrote to Titus, then in Crete, to come to him, Titus iii, 12; but others, with greater probability, are of opinion, that the city of Nicopolis, where St. Paul was, was not that of Epirus, but that of Thrace, on the borders of Macedonia, near the river Nessus. Emmaus in Palestine was also called Nicopolis by the Romans. NIGHT. The ancient Hebrews began their artificial day in the evening, and ended it the next evening; so that the night preceded the day, whence it is said, evening and morning one day,” Gen. i, 5. They allowed twelve hours to the night, and twelve to the day. Night is put for a time of affliction and adversity: Thou hast proved mine heart, thou hast visited me in the night, thou hast tried me,” Psalm xvii, 3; that is, by adversity and tribulation. And the morning cometh, and also the night,” Isaiah xxi, 12. Night is also put for the time of death: The night cometh, wherein no man can work,” John ix, 4. Children of the day, and children of the night, in a moral and figurative sense, denote good men and wicked men, Christians and Gentiles. The disciples of the Son of God are children of light: they belong to the light, they walk in the light of truth; while the children of the night walk in the darkness of ignorance and infidelity, and perform only works of darkness. Ye are all the children of the light, and the children of the day; we are not of the night, nor of darkness,” 1 Thess. v, 5. NIGHT-HAWK, תחמס, Lev. xi, 16; Deut. xiv, 15. That this is a voracious bird seems clear from the import of its name; and interpreters are generally agreed to describe it as flying by night. On the whole, it should seem to be the strix orientalis, which Hasselquist thus describes: It is of the size of the common owl, and lodges in the large buildings or ruins of Egypt and Syria, and sometimes even in the dwelling houses. The Arabs settled in Egypt call it massasa,” and the Syrians banu.” It is extremely voracious in Syria; to such a degree, that if care is not taken to shut the windows at the coming on of night, he enters the houses and kills the children: the women, therefore, are very much afraid of him.
- 64. NILE, the river of Egypt, whose fountain is in the Upper Ethiopia. After having watered several kingdoms, the Nile continues its course far into the kingdom of Goiam. Then it winds about again, from the east to the north. Having crossed several kingdoms and provinces, it falls into Egypt at the cataracts, which are waterfalls over steep rocks of the length of two hundred feet. At the bottom of these rocks the Nile returns to its usual pace, and thus flows through the valley of Egypt. Its channel, according to Villamont, is about a league broad. At eight miles below Grand Cairo, it is divided into two arms, which make a triangle, whose base is at the Mediterranean Sea, and which the Greeks call the Delta, because of its figure Δ. These two arms are divided into others, which discharge themselves into the Mediterranean, the distance of which from the top of the Delta is about twenty leagues. These branches of the Nile the ancients commonly reckoned to be seven. Ptolemy makes them nine, some only four, some eleven, some fourteen. Homer, Xenophon, and Diodorus Siculus testify, that the ancient name of this river was Egyptus; and the latter of these writers says, that it took the name Nilus only since the time of a king of Egypt called by that name. The Greeks gave it the name of Melas; and Diodorus Siculus observes, that the most ancient name by which the Grecians have known the Nile was Oceanus. The Egyptians paid divine honours to this river, and called it Jupiter Nilus. Very little rain ever falls in Egypt, never sufficient to fertilize the land; and but for the provision of this bountiful river, the country would be condemned to perpetual sterility. As it is, from the joint operation of the regularity of the flood, the deposit of mud from the water of the river and the warmth of the climate, it is the most fertile country in the world; the produce exceeding all calculation. It has in consequence been, in all ages, the granary of the east; and has on more than one occasion, an instance of which is recorded in the history of Joseph, saved the neighbouring countries from starvation. It is probable, that, while in these countries, on the occasion referred to, the seven years’ famine was the result of the absence of rain, in Egypt it was brought about by the inundation being withheld: and the consternation of the Egyptians, at
- 65. witnessing this phenomenon for seven successive years, may easily be conceived. The origin and course of the Nile being unknown to the ancients, its stream was held, and is still held by the natives, in the greatest veneration; and its periodical overflow was viewed with mysterious wonder. But both of these are now, from the discoveries of the moderns, better understood. It is now known, that the sources, or permanent springs, of the Nile are situated in the mountains of Abyssinia, and the unexplored regions to the west and south-west of that country; and that the occasional supplies, or causes of the inundation, are the periodical rains which fall in those districts. For a correct knowledge of these facts, and of the true position of the source of that branch of the river, which has generally been considered to be the continuation of the true Nile, we are indebted to our countryman, the intrepid and indefatigable Bruce. Although the Nile, by way of eminence, has been called the river of Egypt,” it must not be confounded with another stream so denominated in Scripture, an insignificant rivulet in comparison, which falls into the Mediterranean below Gaza. NIMROD. He is generally supposed to have been the immediate son of Cush, and the youngest, or sixth, from the Scriptural phrase, Cush begat Nimrod,” after the mention of his five sons, Gen. x, 8. But the phrase is used with considerable latitude, like father” and son,” in Scripture. And the beginning of his kingdom was Babel, and Erech, and Accad, and Calneh, in the land of Shinar: out of that land he went forth to invade Assyria; and built Nineveh, and the city Rehoboth, and Calah, and Resin, between Nineveh and Calah: the same is a great city,” Gen. x, 8–12. Though the main body of the Cushites was miraculously dispersed, and sent by Providence to their destinations along the sea coasts of Asia and Africa, yet Nimrod remained behind, and founded an empire in Babylonia, according to Berosus, by usurping the property of the Arphaxadites in the land of Shinar; where the beginning of his kingdom was Babel,” or Babylon, and other towns: and, not satisfied with this, he next invaded Assur, or Assyria, east of the Tigris, where he built Nineveh, and several other towns. The marginal reading of our English Bible, He went out into Assyria,” or to invade Assyria, is here adopted in preference to
- 66. that in the text: And out of that land went forth Ashur, and builded Nineveh,” &c. The meaning of the word Nineveh may lead us to his original name, Nin, signifying a son,” the most celebrated of the sons of Cush. That of Nimrod, or Rebel,” was probably a parody, or nickname, given him by the oppressed Shemites, of which we have several instances in Scripture. Thus nahash, the brazen serpent” in the wilderness, was called by Hezekiah, in contempt, nehushtan, a piece of brass,” when he broke it in pieces, because it was perverted into an object of idolatrous worship by the Jews, 2 Kings xviii, 4. Nimrod, that arch rebel, who first subverted the patriarchal government, introduced also the Zabian idolatry, or worship of the heavenly host; and, after his death, was deified by his subjects, and supposed to be translated into the constellations of Orion, attended by his hounds, Sirius and Canicula, and still pursuing his favourite game, the great bear; supposed also to be translated into ursa major, near the north pole; as admirably described by Homer,--
- 67. Ἄρκτον θ’, ἣν καὶ ἄμαξαν ἐπίκλησιν καλέουσιν, Ἤ τ’ αὐτου ϛρέφεται, καὶ τ’ Ὠρίωνα δοκεύει. Iliad xviii, 485. And the bear, surnamed also the wain, by the Egyptians, who is turning herself about there, and watching Orion.” Homer also introduces the shade of Orion, as hunting in the Elysian fields,-- Τὸν δὲ μέτ’, Ὠρίωνα πελώριον είσενόησα Θῆρας ὁμοῦ εἰλεῦντα, κατ’ ἀσφοδελον λειμωνα· Τοὺς αὐτος κατέπεφνεν ἐν οἰοπόλοισιν ὄρεσσι Χερσὶν ἔχων ρὅπαλον παγχάλκεον, αἰὲν ἀαγές. Odyss. xi, 571. Next, I observed the mighty Orion Chasing wild beasts through an asphodel mead, Which himself had slain on the solitary mountains: Holding in his hands a solid brazen mace, ever unbroken.” The Grecian name of this mighty hunter” may furnish a satisfactory clue to the name given him by the impious adulation of the Babylonians and Assyrians. Ὠρίων nearly resembles, Ὀυρίαν, the oblique case of Ὀυρίας, which is the Septuagint rendering of Uriah, a proper name in Scripture, 2 Sam. xi, 6–21. But Uriah, signifying the light of the Lord,” was an appropriate appellation of that most brilliant constellation. He was also called Baal, Beel, Bel, or Belus, signifying lord,” or master,” by the Phenicians, Assyrians, and Greeks; and Bala Rama, by the Hindus. At a village called Bala-deva, or Baldeo in the vulgar dialect, thirteen miles east by south from Muttra, in Hindustan, there is a very ancient statue of Bala Rama, in which he is represented with a ploughshare in his left hand, and a thick cudgel in his right, and his shoulders covered with the skin of a tiger. Captain Wilford supposes that the ploughshare was designed to hook his enemies; but may it not more naturally denote the constellation of the great bear, which strikingly represents the figure
- 68. of a plough in its seven bright stars; and was probably so denominated by the earliest astronomers, before the introduction of the Zabian idolatry, as a celestial symbol of agriculture? The thick cudgel corresponds to the brazen mace of Homer. And it is highly probable that the Assyrian Nimrod, or Hindu Bala, was also the prototype of the Grecian Hercules, with his club and lion’s skin. Nimrod is said to have been a mighty hunter before the Lord;” which the Jerusalem paraphrast interprets of a sinful hunting after the sons of men to turn them off from the true religion. But it may as well be taken in a more literal sense, for hunting of wild beasts; inasmuch as the circumstance of his being a mighty hunter is mentioned with great propriety to introduce the account of his setting up his kingdom; the exercise of hunting being looked upon in ancient times as a means of acquiring the rudiments of war; for which reason, the principal heroes of Heathen antiquity, as Theseus, Nestor, &c, were, as Xenophon tells us, bred up to hunting. Beside, it may be supposed, that by this practice Nimrod drew together a great company of robust young men to attend him in his sport, and by that means increased his power. And by destroying the wild beasts, which, in the comparatively defenceless state of society in those early ages, were no doubt very dangerous enemies, he might, perhaps, render himself farther popular; thereby engaging numbers to join with him, and to promote his chief design of subduing men, and making himself master of many nations. NINEVEH. This capital of the Assyrian empire could boast of the remotest antiquity. Tacitus styles it, Vetustissima sedes Assyriæ;” [the most ancient seat of Assyria;] and Scripture informs us that Nimrod, after he had built Babel, in the land of Shinar, invaded Assyria, where he built Nineveh, and several other cities, Genesis x, 11. Its name denotes the habitation of Nin,” which seems to have been the proper name of that rebel,” as Nimrod signifies. And it is uniformly styled by Herodotus, Xenophon, Diodorus, Lucian, &c, Ἡ Νίνος, the city of Ninus.” And the village of Nunia, opposite Mosul, in its name, and the tradition of the natives, ascertains the site of the ancient city, which was near the castle of Arbela, according to Tacitus, so celebrated for the decisive victory of Alexander the Great
- 69. over the Persians there; the site of which is ascertained by the village of Arbil, about ten German miles to the east of Nunia, according to Niebuhr’s map. Nineveh at first seems only to have been a small city, and less than Resen, in its neighbourhood; which is conjectured by Bochart, and not without reason, to have been the same as Larissa, which Xenophon describes as the ruins of a great city, formerly inhabited by the Medes,” and which the natives might have described as belonging la Resen, to Resen.” Nineveh did not rise to greatness for many ages after, until its second founder, Ninus II., about B. C. 1230, enlarged and made it the greatest city in the world. According to Diodorus, it was of an oblong form, a hundred and fifty stadia long, and ninety broad, and, consequently, four hundred and eighty in circuit, or forty-eight miles, reckoning ten stadia to an English mile, with Major Rennel. And its walls were a hundred feet high, and so broad that three chariots could drive on them abreast; and on the walls were fifteen hundred towers, each two hundred feet high. We are not, however, to imagine that all this vast enclosure was built upon: it contained great parks and extensive fields, and detached houses and buildings, like Babylon, and other great cities of the east even at the present day, as Bussorah, &c. And this entirely corresponds with the representations of Scripture. In the days of the Prophet Jonah, about B. C. 800, it seems to have been a “great city, an exceeding great city, of three days’ journey,” Jonah i, 2; iii, 3; perhaps in circuit. The population of Nineveh, also, at that time was very great. It contained more than sixscore thousand persons that could not discern between their right hand and their left, beside much cattle,” Jonah iv, 11. Reckoning the persons to have been infants of two years old and under, and that these were a fifth part of the whole, according to Bochart, the whole population would amount to six hundred thousand souls. The same number Pliny assigns for the population of Seleucia, on the decline of Babylon. This population shows that a great part of the city must have been left open and unbuilt. The threatened overthrow of Nineveh within three days, was, by the general repentance and humiliation of the inhabitants, from the highest to the lowest, suspended for near two hundred years, until
- 70. their iniquity came to the full;” and then the prophecy was literally accomplished, in the third year of the siege of the city, by the combined Medes and Babylonians; the king, Sardanapalus, being encouraged to hold out in consequence of an ancient prophecy, that Nineveh should never be taken by assault, till the river became its enemy; when a mighty inundation of the river, swollen by continual rains, came up against a part of the city, and threw down twenty stadia of the wall in length; upon which, the king, conceiving that the oracle was accomplished, burned himself, his concubines, eunuchs, and treasures; and the enemy, entering by the breach, sacked and rased the city, about B. C. 606. Diodorus, also, relates that Belesis, the governor of Babylon, obtained from Arbaces, the king of Media, the ashes of the palace, to erect a mount with them near the temple of Belus at Babylon; and that he forthwith prepared shipping, and, together with the ashes, carried away most of the gold and silver, of which he had private information given him by one of the eunuchs who escaped the fire. Dr. Gillies thinks it incredible that these could be transported from Nineveh to Babylon, three hundred miles distant; but likely enough, if Nineveh was only fifty miles from Babylon, with a large canal of communication between them, the Nahar Malka, or Royal River. But we learn from Niebuhr, that the conveyance of goods from Nosul to Bagdat by the Tigris is very commodious, in the very large boats called helleks; in which, in spring, when the river is rapid, the voyage may be made in three or four days, which would take fifteen by land. The complete demolition of such immense piles as the walls and towers of Nineveh may seem matter of surprise to those who do not consider the nature of the materials of which they were constructed, that is, of bricks, dried or baked in the sun, and cemented with bitumen, which were apt to be dissolved” by water, or to moulder away by the injuries of the weather. Beside, in the east, the materials of ancient cities have been often employed in the building of new ones in the neighbourhood. Thus Mosul was built with the spoils of Nineveh. Tauk Kesra, or the Palace of Chosroes, appears to have been built of bricks brought from the ruins of Babylon; and so was Hellah, as the dimensions are nearly the same, and the proportions so singular.
- 71. And when such materials could conveniently be transported by inland navigations, they are to be found at very great distances from their ancient place, much farther, indeed, than are Bagdat and Seleucia, or Ctesiphon, from Babylon. The book of Nahum was avowedly prophetic of the destruction of Nineveh; and it is there foretold that the gates of the river shall be opened, and the palace shall be dissolved. Nineveh of old, like a pool of water, with an overflowing flood he will make an utter end of the place thereof,” Nahum ii, 6; i, 8, 9. The historian describes the facts by which the other predictions of the prophet were as literally fulfilled. He relates that the king of Assyria, elated with his former victories, and ignorant of the revolt of the Bactrians, had abandoned himself to scandalous inaction; had appointed a time of festivity, and supplied his soldiers with abundance of wine; and that the general of the enemy, apprised by deserters, of their negligence and drunkenness, attacked the Assyrian army while the whole of them were fearlessly giving way to indulgence, destroyed great part of them, and drove the rest into the city. The words of the prophet were hereby verified: While they be folden together as thorns, and while they are drunken as drunkards, they shall be devoured as stubble fully dry,” Nahum i, 10. The prophet promised much spoil to the enemy: Take the spoil of silver, take the spoil of gold; for there is no end of the store and glory out of all the pleasant furniture,” Nahum ii, 9. And the historian affirms that many talents of gold and silver, preserved from the fire, were carried to Ecbatana. According to Nahum, iii, 15, the city was not only to be destroyed by an overflowing flood, but the fire, also, was to devour it; and, as Diodorus relates, partly by water, partly by fire, it was destroyed. The utter and perpetual destruction and desolation of Nineveh were foretold: The Lord will make an utter end of the place thereof. Affliction shall not rise up the second time. She is empty, void, and waste,” Nahum i, 8, 9; ii, 10; iii, 17–19. The Lord will stretch out his hand against the north, and destroy Assyria, and will make Nineveh a desolation, and dry like a wilderness. How is she become a desolation, a place for beasts to lie down in,” Zeph. ii, 13–15. In the second century, Lucian, a native of a city on the banks of the
- 72. Euphrates, testified that Nineveh was utterly perished, that there was no vestige of it remaining, and that none could tell where once it was situated. This testimony of Lucian, and the lapse of many ages during which the place was not known where it stood, render it at least somewhat doubtful whether the remains of an ancient city, opposite to Mosul, which have been described as such by travellers, be indeed those of ancient Nineveh. It is, perhaps, probable that they are the remains of the city which succeeded Nineveh, or of a Persian city of the same name, which was built on the banks of the Tigris by the Persians subsequently to A. D. 230, and demolished by the Saracens, A. D. 632. In contrasting the then existing great and increasing population, and the accumulating wealth of the proud inhabitants of the mighty Nineveh, with the utter ruin that awaited it, the word of God by the Prophet Nahum, was, Make thyself many as the canker worm, make thyself many as the locusts. Thou hast multiplied thy merchants above the stars of heaven: the canker worm spoileth and flieth away. Thy crowned are as the locusts, and thy captains as the great grasshoppers which camp in the hedges in the cold day: but when the sun riseth, they flee away; and their place is not known where they are,” or were. Whether these words imply that even the site of Nineveh would in future ages be uncertain or unknown; or, as they rather seem to intimate, that every vestige of the palaces of its monarchs, of the greatness of its nobles, and of the wealth of its numerous merchants, would wholly disappear; the truth of the prediction cannot be invalidated under either interpretation. The avowed ignorance respecting Nineveh, and the oblivion which passed over it, for many an age, conjoined with the meagreness of evidence to identify it, still prove that the place where it stood was long unknown, and that, even now, it can scarcely with certainty be determined. And if the only spot that bears its name, or that can be said to be the place where it was, be indeed the site of one of the most extensive of cities on which the sun ever shone, and which continued for many centuries to be the capital of Assyria,--the principal mounds, few in number, which show neither bricks, stones, nor other materials of building,--but are in many places overgrown with grass, and resemble the mounds left by
- 73. intrenchments and fortifications of ancient Roman camps, and the appearances of other mounds and ruins less marked than even these, extending for ten miles, and widely spread, and seeming to be the wreck of former buildings,--show that Nineveh is left without one monument of royalty, without any token whatever of its splendour or wealth; that their place is not known where they were; and that it is indeed a desolation, empty, void, and waste,” its very ruins perished, and less than the wreck of what it was. Such an utter ruin, in every view, has been made of it; and such is the truth of the divine predictions! NISAN, a month of the Hebrews, answering to our March, and which sometimes takes from February or April, according to the course of the moon. It was made the first month of the sacred year, at the coming out of Egypt, Exod. xii, 2; and it was the seventh month of the civil year. By Moses it is called Abib. The name Nisan was introduced only since the time of Ezra, and the return from the captivity of Babylon. NISROCH, a god of the Assyrians. Sennacherib was killed by two of his sons, while he was paying his adorations in the temple of this deity, 2 Kings xix, 37; Isaiah xxxvii, 38. It is uncertain who this god was. NITRE, נתר, Prov. xxv, 20; Jer. ii, 22. This is not the same that we call nitre, or saltpetre, but a native salt of a different kind, distinguished among naturalists by the name of natrum. The natrum of the ancients was an earthy alkaline salt. It was found in abundance separated from the water of the lake Natron in Egypt. It rises from the bottom of the lake to the top of the water, and is there condensed by the heat of the sun into the hard and dry form in which it is sold. This salt thus scummed off is the same in all respects with the Smyrna soap earth. Pliny, Matthiolus, and Agricola, have described it to us: Hippocrates, Galen, Dioscorides, and others, mention its uses. It is also found in great plenty in Sindy, a province in the inner part of Asia, and in many other parts of the east; and might be had in any quantities. The learned Michaëlis plainly demonstrates, from the nature of the thing and the context, that this fossil and natural alkali must be that which the Hebrews called
- 74. nether. Solomon must mean the same when he compares the effect which unseasonable mirth has upon a man in affliction to the action of vinegar upon nitre, Prov. xxv, 20; for vinegar has no effect upon what we call nitre, but upon the alkali in question has a great effect, making it rise up in bubbles with much effervescence. It is of a soapy nature, and was used to take spots from clothes, and even from the face. Jeremiah alludes to this use of it, ii, 22. NO, or NO-AMMON, a city of Egypt, supposed to be Thebes. NOAH, the son of Lamech. Amidst the general corruption of the human race, Noah only was found righteous, Gen. vi, 9. He therefore found grace in the sight of the Lord,” and was directed for his preservation to make an ark, the shape and dimensions of which were prescribed by the Lord. In A. M. 1656, and in the six hundredth year of his age, Noah, by divine appointment, entered the ark with his family, and all the animals collected for the renewal of the world. (See Deluge.) After the ark had stranded, and the earth was in a measure dried, Noah offered a burnt-sacrifice to the Lord, of the pure animals that were in the ark; and the Lord was pleased to accept of his offering, and to give him assurance that he would no more destroy the world by water, Genesis ix. He gave Noah power over all the brute creation, and permitted him to kill and eat of them, as of the herbs and fruits of the earth, except the blood, the use of which was prohibited. After the deluge Noah lived three hundred and fifty years; and the whole time of his life having been nine hundred and fifty years, he died, A. M. 2006. According to common opinion, he divided the earth among his three sons, Shem, Ham, and Japheth. To Shem he gave Asia, to Ham Africa, and to Japheth Europe. Some will have it, that beside these three sons he had several others. St. Peter calls Noah a preacher of righteousness, because before the deluge he was incessantly preaching and declaring to men, not only by his discourses, but by the building of the ark, in which he was employed a hundred and twenty years, that the cloud of divine vengeance was about to burst upon them. But his faithful ministry produced no effect, since, when the deluge came, it found mankind practising their usual enormities, Matt. xxiv, 37. Several learned men have observed that the Heathens confounded
- 75. Saturn, Deucalion, Ogyges, the god Cœlus or Ouranus, Janus, Protheus, Prometheus, &c, with Noah. The fable of Deucalion and his wife Pyrrha is manifestly drawn from the history of Noah. The rabbins pretend that God gave Noah and his sons certain general precepts, which contain, according to them, the natural duties which are common to all men indifferently, and the observation of which alone will be sufficient to save them. After the law of Moses was given, the Hebrews would not suffer any stranger to dwell in their country, unless he would conform to the precepts of Noah. In war, they put to death without quarter all who were ignorant of them. These precepts are seven in number: the first was against the worship of idols; the second, against blasphemy, and required to bless the name of God; the third, against murder; the fourth, against incest and all uncleanness; the fifth, against theft and rapine; the sixth required the administration of justice; the seventh was against eating flesh with life. But the antiquity of these precepts is doubted, since no mention of them is made in the Scripture, or in the writings of Josephus, or in Philo; and none of the ancient fathers knew any thing of them. NOD, Land of, the country to which Cain withdrew after the murder of Abel. As the precise situation of this country cannot possibly be known, so it has given rise to much ingenious speculation. All that we are told of it is, that it was on the east of Eden,” or, as it may be rendered, before Eden;” which very country of Eden is no sure guide for us, as the situation of that also is disputed. But, be it on the higher or lower Euphrates, (see Eden,) the land of Nod which stood before it with respect to the place where Moses wrote, may still preserve the curse of barrenness passed on it for Cain’s sake, namely, in the deserts of Syria or Arabia. The Chaldee interpreters render the word Nod, not as the proper name of a country, but as an appellative applied to Cain himself, signifying a vagabond or fugitive, and read, He dwelt a fugitive in the land.” But the Hebrew reads expressly, He dwelt in the land of Nod.” NONCONFORMISTS, dissenters from the church of England; but the term applies more particularly to those ministers who were
- 76. ejected from their livings by the Act of Uniformity in 1662; the number of whom, according to Dr. Calamy, was nearly two thousand; and to the laity who adhered to them. The celebrated Mr. Locke says, Bartholomew-day (the day fixed by the Act of Uniformity) was fatal to our church and religion, by throwing out a very great number of worthy, learned, pious, and orthodox divines, who could not come up to this and other things in that act. And it is worth your knowledge, that so great was the zeal in carrying on this church affair, and so blind was the obedience required, that if you compare the time of passing the act with the time allowed for the clergy to subscribe the book of Common Prayer thereby established, you shall plainly find, it could not be printed and distributed, so as one man in forty could have seen and read the book before they did so perfectly assent and consent thereto.” By this act, the clergy were required to subscribe, ex animo, [sincerely,] their assent and consent to all and every thing contained in the book of Common Prayer,” which had never before been insisted on, so rigidly as to deprive them of their livings and livelihood. Several other acts were passed about this time, very oppressive both to the clergy and laity. In the preceding year 1661, the Corporation Act incapacitated all persons from offices of trust and honour in a corporation, who did not receive the sacrament in the established church. The Conventicle Act, in 1663 and 1670, forbade the attendance at conventicles; that is, at places of worship other than the establishment, where more than five adults were present beside the resident family; and that under penalties of fine and imprisonment by the sentence of magistrates without a jury. The Oxford Act of 1665 banished nonconforming ministers five miles from any corporate town sending members to parliament, and prohibited them from keeping or teaching schools. The Test Act of the same year required all persons, accepting any office under government, to receive the sacrament in the established church. Such were the dreadful consequences of this intolerant spirit, that it is supposed that near eight thousand died in prison in the reign of Charles II. It is said that Mr. Jeremiah White had carefully collected a list of those who had suffered between Charles II. and the
- 77. revolution, which amounted to sixty thousand. The same persecutions were carried on in Scotland; and there, as well as in England, numbers, to avoid the persecution, left their country. But, notwithstanding all these dreadful and furious attacks upon the dissenters, they were not extirpated. Their very persecution was in their favour. The infamous character of their informers and persecutors; their piety, zeal, and fortitude, no doubt, had influence on considerate minds; and, indeed, they had additions from the established church, which several clergymen in this reign deserted as a persecuting church, and took their lot among them. King William coming to the throne, the famous Toleration Act passed, by which they were exempted from suffering the penalties above mentioned, and permission was given them to worship God according to the dictates of their own consciences. In the reign of George III., the Act for the Protection of Religious Worship superseded the Act of Toleration, by still more liberal provisions in favour of religious liberty; and in the last reign the Test and Corporation Acts were repealed. NOPH, Memphis, a celebrated city of Egypt, and, till the time of the Ptolemies, who removed to Alexandria, the residence of the ancient kings of Egypt. It stood above the dividing of the river Nile, where the Delta begins. Toward the south of this city stood the famous pyramids, two of which were esteemed the wonders of the world; and in this city was fed the ox Apis, which Cambyses slew, in contempt of the Egyptians who worshipped it as a deity. The kings of Egypt took much pleasure in adorning this city; and it continued in all its beauty till the Arabians made a conquest of Egypt under the Caliph Omar. The general who took it built another city near it, named Fustal, merely because his tent had been a long time set up in that place; and the Fatimite caliphs, when they became masters of Egypt, added another to it, which is known to us at this day by the name of Grand Cairo. This occasioned the utter decay of Memphis, and led to the fulfilment of the prophecy, that it should be waste and without inhabitant.” The prophets often speak of this city, and foretel the miseries it was to suffer from the kings of Chaldea
- 78. and Persia, Isaiah xix, 13; Jer. xliv, 1; xlvi, 14, 19; Hosea ix, 6; Ezek. xxx, 13, 16. NOVATIANS, the followers of Novatian, a priest of Rome, and of Novatus, a priest of Carthage, in the third century. They were distinguished merely by their discipline; for their religious and doctrinal tenets do not appear to be at all different from those of the church. They condemned second marriages, and for ever excluded from their communion all those who after baptism had fallen into sin. They affected very superior purity; and, though they conceived that the worst might possibly hope for eternal life, they absolutely refused to reädmit into their communion any who had lapsed into sin. They separated from the church of Rome, because the members of it admitted into their communion many who had, during a season of persecution, rejected the Christian faith. NUMBERS, a canonical book of the Old Testament, being the fourth of the Pentateuch, or five books of Moses; and receives its denomination from the numbering of the families of Israel by Moses and Aaron, who mustered the tribes, and marshalled the army, of the Hebrews in their passage through the wilderness. A great part of this book is historical, relating several remarkable events which happened in that journey, and also mentioning various of their journeyings in the wilderness. This book comprehends the history of about thirty-eight years, though the greater part of the things recorded fell out in the first and last of those years; and it does not appear when those things were done which are recorded in the middle of the book. See Pentateuch. NURSE. The nurse in an eastern family is always an important personage. Modern travellers inform us, that in Syria she is considered as a sort of second parent, whether she has been foster- mother or otherwise. She always accompanies the bride to her husband’s house, and ever remains there an honoured character. Thus it was in ancient Greece. This will serve to explain Genesis xxiv, 59: And they sent away Rebekah their sister, and her nurse.” In Hindostan the nurse is not looked upon as a stranger, but becomes one of the family, and passes the remainder of her life in the midst of the children she has suckled, by whom she is honoured and
- 79. cherished as a second mother. In many parts of Hindostan are mosques and mausoleums, built by the Mohammedan princes, near the sepulchres of their nurses. They are excited by a grateful affection to erect these structures in memory of those who with maternal anxiety watched over their helpless infancy: thus it has been from time immemorial. OAK. The religious veneration paid to this tree, by the original natives of our island in the time of the Druids, is well known to every reader of British history. We have reason to think that this veneration was brought from the east; and that the Druids did no more than transfer the sentiments their progenitors had received in oriental countries. It should appear that the Patriarch Abraham resided under an oak, or a grove of oaks, which our translators render the plain of Mamre; and that he planted a grove of this tree, Gen. xiii, 18. In fact, since in hot countries nothing is more desirable than shade, nothing more refreshing than the shade of a tree, we may easily suppose the inhabitants would resort for such enjoyment to Where’er the oak’s thick branches spread A deeper, darker shade. Oaks, and groves of oaks, were esteemed proper places for religious services; altars were set up under them, Joshua xxiv, 26; and, probably, in the east as well as in the west, appointments to meet at conspicuous oaks were made, and many affairs were transacted or treated of under their shade, as we read in Homer, Theocritus and other poets. It was common among the Hebrews to sit under oaks, Judges vi, 11; 1 Kings xiii, 14. Jacob buried idolatrous images under an oak, Gen. xxxv, 4; and Deborah, Rebekah’s nurse, was buried under one of these trees, Genesis xxxv, 8. See 1 Chron. x, 12. Abimelech was made king under an oak, Judges ix, 6. Idolatry was practised under oaks, Isaiah i, 29; lvii, 5; Hosea iv, 13. Idols were made of oaks, Isa. xliv, 14.
- 80. OATH, a solemn invocation of a superior power, admitted to be acquainted with all the secrets of our hearts, with our inward thoughts as well as our outward actions, to witness the truth of what we assert, and to inflict his vengeance upon us if we assert what is not true, or promise what we do not mean to perform. Almost all nations, whether savage or civilized, whether enjoying the light of revelation or led only by the light of reason, knowing the importance of truth, and willing to obtain a barrier against falsehood, have had recourse to oaths, by which they have endeavoured to make men fearful of uttering lies, under the dread of an avenging Deity. Among Christians, an oath is a solemn appeal for the truth of our assertions, the sincerity of our promises, and the fidelity of our engagements, to the one only God, the Judge of the whole earth, who is every where present, and sees, and hears, and knows, whatever is said, or done, or thought in any part of the world. Such is that Being whom Christians, when they take an oath, invoke to bear testimony to the truth of their words, and the integrity of their hearts. Surely, then, if oaths be a matter of so much moment, it well behoves us not to treat them with levity, nor ever to take them without due consideration. Hence we ought, with the utmost vigilance, to abstain from mingling oaths in our ordinary discourse, and from associating the name of God with low or disgusting images, or using it on trivial occasions, as not only a profane levity in itself, but tending to destroy that reverence for the supreme Majesty which ought to prevail in society, and to dwell in our own hearts. The forms of oaths,” says Dr. Paley, like other religious ceremonies, have in all ages been various; consisting, however, for the most part of some bodily action, and of a prescribed form of words.” Among the Jews, the juror held up his right hand toward heaven, Psalm cxliv, 8; Rev. x, 5. The same form is retained in Scotland still. Among the Jews, also, an oath of fidelity was taken by the servant’s putting his hand under the thigh of his lord, Genesis xxiv, 2. Among the Greeks and Romans, the form varied with the subject and occasion of the oath: in private contracts, the parties took hold of each other’s hands, while they swore to the performance; or they touched the altar of the god by whose divinity
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