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Chapter 07 - Revenue and Collection Cycle
7-1
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
CHAPTER 07
Revenue and Collection Cycle
LEARNING OBJECTIVES
Review
Checkpoints
Multiple
Choice
Exercises,
Problems, and
Simulations
1. Discuss inherent risks related to the
revenue and collection cycle with a
focus on improper revenue
recognition.
1, 2, 3 29 62, 68, 72
2. Describe the revenue and collection
cycle, including typical source
documents and controls procedures.
4, 5, 6, 7, 8 31, 32, 39, 43,
46, 52
60, 67, 70
3. Give examples of tests of controls
over customer credit approval,
delivery, and recording of accounts
receivable.
9, 10, 11, 12,
13, 14
30, 33, 34, 35,
37, 47, 51, 55
61, 62
4. Give examples of substantive
procedures in the revenue and
collection cycle and relate them to
assertions about account balances at
the end of the period.
15, 16, 17, 18,
19, 20, 21, 22
36, 38, 41, 42,
44, 45, 48, 49,
50, 53, 56, 57,
58, 59
63, 64, 66, 73,
74, 75
5. Describe some common errors and
frauds in the revenue and collection
cycle and design some audit and
investigation procedures for detecting
them.
23, 24, 25 26,
27, 28
40, 54 65, 69, 71

7-2
SOLUTIONS FOR REVIEW CHECKPOINTS
7.1 Revenue recognition refers to including revenue in the financial statements. According to GAAP, this is
done when revenues are (1) realized or realizable and (2) earned.
7.2 Revenue recognition is used as a primary means for inflating profits for several reasons. First, it is not
always straightforward when revenues have been earned. Sales can be structured with return provisions or
can have other performance provisions attached. Second, the timing of shipments at year-end may be easy
to falsify. Third, markets often value companies based on a multiple of its revenue instead of net income.
7.3 New companies often do not show a profit during their first few years. Therefore, creditors and investors
often place more emphasis on the revenues, especially looking for revenue growth that might lead to future
profitability. Knowing this, management could try to inflate revenues.
7.4 The basic sequence of activities and accounting in a revenue and collection cycle is:
a. Receiving and processing customer orders. Entering data in an order system and obtaining a credit
check.
b. Delivering goods and services to customers. Authorizing release from storekeeping to shipping to
customer. Entering shipping information in the accounting system.
c. Billing customers, producing sales invoices. Accounting for accounts receivable.
d. Collecting cash and depositing it in the bank. Accounting for cash receipts.
e. Reconciling bank statements.
7.5 When documents such as sales orders, shipping documents, and sales invoices are prenumbered, someone
can later account for the numerical sequence and determine whether any transactions have failed to be
recorded. (Completeness assertion.)
7.6 Access to computer terminals should be restricted so that only authorized persons can enter or change
transaction data. Access to master files is important because changes in them affect automatic computer
controls, such as credit checking and accurate inventory pricing.
7.7 Auditors could examine these files for evidence of:
• Unrecorded sales — pending order master file,
• Inadequate credit checks — credit data/check files
• Incorrect product unit prices — price list master file
7.8 With a sample of customer accounts receivable:
• Find the support for debit entries in the sales journal file. Expect to find evidence (copy) of a sales invoice,
shipping document, and customer order. The sales invoice indicates the shipping date.
• Find the support for credit entries in the cash receipts journal file. Expect to find a remittance advice (entry
on list), which corresponds to detail on a deposit slip, on a deposit actually in a bank statement for the day
posted in the customers’ accounts.

7-3
7.9 The account balances in a revenue and collection cycle include:
• Cash in bank
• Accounts receivable
• Allowance for doubtful accounts
• Bad debt expense
• Sales revenue
• Sales returns
• Sales allowances
• Sales discounts
7.10 These specific control procedures (in addition to separation of duties and responsibilities) should be in
place and operating in a control system governing revenue recognition and cash accounting:
• No sales order should be entered without a customer order.
• A credit-check code or manual signature should be recorded by an authorized person.
• Access to inventory and the shipping area should be restricted to authorized persons.
• Access to billing terminals and blank invoice forms should be restricted to authorized personnel.
• Accountants should be instructed to record sales and accounts receivable when all the supporting
documentation of shipment is in order, and care should be taken to record sales and receivables as of the
date goods and services were shipped, and cash receipts on the date the payments are received.
• Customer invoices should be compared with bills of lading and customer orders to assure that the customer
is sent the goods ordered at the proper location for the proper prices and that the quantity being billed is the
same as the quantity shipped
• Pending order files should be reviewed in a timely manner to avoid failure to bill the customer and record
shipments
• Bank statements should be reconciled in detail monthly.
7.11 The purpose of the walkthrough is to obtain an understanding of the transaction flow, the control
procedures, and the populations of documents that may be utilized in tests of controls. In a walkthrough of
a sales transaction, auditors take a small sample (usually 1–3 items) of a sales transaction and trace it from
the initial customer order through credit approval, billing, and delivery of goods to the entry in the sales
journal and subsidiary accounts receivable records, and then its subsequent collection and cash deposit.
Sample documents are collected, and employees in each department are questioned about their specific
duties. The information gained from documents and employees can be compared to answers obtained on an
internal control questionnaire to ensure proper Procedures are taking place.
7.12 The assertions made about classes of transactions and events in the revenue and collection cycle are:
• Sales and related events that have been recorded have occurred and pertain to the entity.
• All sales and related events that should have been recorded have been recorded.
• Amounts and other data related to sales transactions and events have been recorded properly.
• Sales and related events have been recorded in the correct period.
• Sales and related events have been recorded in the proper accounts.
7.13 In general, the actions in tests of controls involve vouching, tracing, observing, scanning, and recalculating.

7-4
7.14 Dual direction tests of controls sampling refers to procedures that test file contents in two “directions”: the
occurrence direction and the completeness direction. The occurrence direction involves a sample from the
account balance (e.g., sales revenue) vouched to supporting sales and shipping documents for evidence of
occurrence. The completeness direction is a sample from the population that represents all sales (e.g.,
shipping document files) traced to the sales journal or sales account for evidence that no transactions
(shipments, sales) were omitted.
7.15 It is important to place emphasis on the existence assertion because auditors have often been sued for
malpractice by providing unqualified reports on financial statements that overstated assets and revenues
and understated expenses. For example, credit sales recorded too early (e.g., a fictitious sale) result in
overstated accounts receivable and overstated sales revenue.
7.16 These procedures are usually the most useful for auditing the existence assertion:
Confirmation. Letters of confirmation asking for a report of the balances owed to the company can be sent
to customers.
Verbal Inquiry. Inquiries to management usually do not provide very convincing evidence about existence
and ownership. However, inquiries about the company’s agreements to pledge or sell with recourse
accounts receivable in connection with financings should always be made.
Examination of Documents (vouching). Evidence of existence can be obtained by examining shipping
documents. Examination of loan documents may yield evidence of the need to disclose receivables pledged
as loan collateral.
Scanning. Assets are supposed to have debit balances. A computer can be used to scan large files of
accounts receivable, inventory, and fixed assets for uncharacteristic credit balances. The names of debtors
can be scanned for officers, directors, and related parties, amounts for which need to be reported separately
or disclosed in the financial statements.
Analytical Procedures. Comparisons of asset and revenue balances with recent history might help detect
overstatements. Relationships such as receivables turnover, gross margin ratio, and sales/asset ratios can be
compared to historical data and industry statistics for evidence of overall reasonableness. Account
interrelationships also can be used in analytical review. For example, sales returns and allowances and sales
commissions generally vary directly with dollar sales volume, bad debt expense usually varies directly with
credit sales volume, and freight expense varies with the physical sales volume. Accounts receivable
write-offs should be compared with earlier estimates of doubtful accounts.
7.17 Comparison of sales and accounts receivable to previous periods provides information about existence.
Other useful analytical procedures include receivables turnover and days of sales in receivables, aging,
gross margin ratio, and sales/asset ratios, which can be compared to historical data and industry statistics
for evidence of overall reasonableness. Auditors may also compare sales to nonfinancial data such as units
sold, number of customers, sales commissions, and so on. These comparisons can be made by product,
period, geographic region, or salesperson.
7.18 A positive confirmation is a request for a response from an independent party whom the auditor has reason
to expect is able to reply. A negative confirmation is a request for a response from the independent party
only if the information is disputed. Negative confirmations should be sent only if the recipient can be
expected to detect an error and reply accordingly. They are normally used for accounts with small balances
when control risk is low.
7.19 Justifications for the decision not to use confirmations for trade accounts receivable in a particular audit
include (a) receivables are not material, (b) confirmations would be ineffective based on prior years’
experience or knowledge that responses could be unreliable, and (c) analytical procedures and other
substantive procedures provide sufficient, competent evidence.

7-5
7.20 Auditors need to take special care in examining sources of accounts receivable confirmation responses.
Auditors need to control the confirmations, including the addresses to which they are sent. History is full
of cases in which confirmations were mailed to company accomplices who had provided false responses.
The auditors should carefully consider features of the reply such as postmarks, FAX, and email responses,
letterhead, electronic mail, telephone number, or other characteristics that may give clues to indicate false
responses. Auditors should follow up electronic and telephone responses to determine their origin (for
example, returning the telephone call to a known number, looking up telephone numbers to determine
addresses, or using a crisscross directory to determine the location of a respondent).
7.21 When positive confirmations are not returned, the auditor should perform the following procedures:
a. Send second and even third requests.
b. Apply subsequent cash receipts.
c. Examine sales orders, invoices, and shipping documents.
d. Examine correspondence files for past due accounts.
7.22 To determine the adequacy of the allowance for doubtful accounts, the auditor reviews subsequent cash
receipts from the customer, discusses unpaid accounts with the credit manager, and examines the credit
files. These should contain the customer’s financial statements, credit reports, and correspondence between
the client and the customer. Based on this evidence, the auditor estimates the likely amount of nonpayment
for the customer, which is included in the estimate of the allowance for doubtful accounts. In addition, an
allowance should be estimated for all other customers, perhaps as a percentage of the current accounts and
a higher percentage of past due accounts. The auditor compares his or her estimate to the balance in the
allowance account and proposes an adjusting entry for the difference.
7.23 Dual-direction testing involves selecting samples to obtain evidence about control over completeness in one
direction and control over occurrence in the other direction. The completeness direction determines
whether all transactions that occurred were recorded (none omitted), and the occurrence direction
determines whether recorded transactions actually occurred (were valid). An example of the completeness
direction is the examination of a sample of shipping documents (from the file of all shipping documents) to
determine whether invoices were prepared and recorded. An example of the occurrence direction is the
examination of a sample of sales invoices (from the file representing all recorded sales) to determine
whether supporting shipping documents exist to verify the fact of an actual shipment. The content of each
file is compared with the other.
7.24 In the Canny Cashier case, if someone other than the assistant controller had reconciled the bank statement
and compared the details of bank deposit slips to cash remittance reports, the discrepancies could have been
noted and followed up. The discrepancies were that customers and amounts on the bank deposit slips to
cash remittance reports did not match.
7.25 To prevent the cash receipts journal and recorded cash sales from reflecting more than the amount shown
on the daily deposit slip, internal controls should ensure that receipts are recorded daily and are complete.
A careful bank reconciliation by an independent person may detect such errors.
7.26 Confirmations to taxpayers who had actually paid their taxes would have produced exceptions, complaints,
and people with their counter receipts. These results would have revealed the embezzlement.
7.27 Auditors might have obtained the following information:
Inquiries: Personnel admitting the practices of backdating shipping documents in a “bill-and-hold” tactic or
personnel describing the 60-day wait for a special journal entry to record customer discounts taken.
Tests of Controls: The sample of customer payment cash receipts would have shown no discount
calculations and authorizations, leading to inquiries about the manner and timing of recording the
discounts.

7-6
Observation: When observing the physical inventory-taking, special notice should be taken of any goods
on the premises but excluded from the inventory. These are often signs of sales recorded too early.
Confirmations of Accounts Receivable: Customers who had not yet been given credit for their discounts can
be expected to take exception to a balance that is too high.
7.28 The auditors would have known about the normal Friday closing of the books for weekly management
reports, and they could have been alerted to the possibility that the accounting employees overlooked the
once-a-year occurrence of the year-end date during the week.
SOLUTIONS FOR MULTIPLE CHOICE QUESTIONS
7.29 a. Incorrect Allowances can be made for anticipated returns if the earning process is
substantially complete.
b. Correct The earning process is complete at this point.
c. Incorrect Under accrual accounting, the cash does not have to be collected, only
collectible
d. Incorrect This is usually the method for determining (b.), but the shipment might be FOB
destination
7.30 a. Incorrect This only initiates the earnings process but it doesn’t complete it.
b. Incorrect This is often the case, but it depends on shipping terms.
c. Correct This is often the same as the bill of lading date.
d. Incorrect Under accrual accounting, the company doesn’t have to wait for the check to
record revenue.
7.31 a. Incorrect This would not have the outstanding balance; however, there are some times
when the auditor confirms the sale instead of the amount receivable.
b. Correct This would have the balance for confirming
c. Incorrect This would not have the individual customer balance
d. Incorrect This would not have the balance outstanding
7.32 a. Incorrect This is an essential part of the cycle.
b. Incorrect This is an essential part of the cycle.
c. Incorrect Cash is affected by the collections.
d. Correct Even though this involves shipments, it is considered part of the expenditure and
disbursement cycle.
7.33 a. Incorrect The sale could occur but not be approved for credit.
b. Incorrect The approval is unrelated to the completeness assertion.
c. Correct Credit approval helps ensure that the sale will be collectible.
d. Incorrect Credit approval will not affect in which period the revenue is earned.
7.34 a. Incorrect The general ledger bookkeeper doesn’t have access to the customer accounts.
b. Incorrect There’s no advantage to separating access to checks and currency.
c. Correct The cash is not in the same physical place as the empployees; therefore it cannot
be stolen.
d. Incorrect Normally checks are made payable to company. That doesn’t prevent lapping.
7.35 a. Correct Impropriety of write-offs can be controlled by the review and approval of
someone outside the credit department.
b. Incorrect Even write-offs of old receivables can conceal a cash shortage.
c. Incorrect The cashier could be the cause of the shortage.
d. Incorrect Write-offs should be separated from the sales function.

7-7
7.36 a. Incorrect This would increase gross profit.
b. Correct Less sales revenue and correct amount of cost of goods sold results in less gross
profit, therefore the ratio of gross profit to sales will decrease. (Actually, the
gross profit numerator will decrease at a greater rate than the sales denominator
in the ratio, causing the ratio to decrease.)
c. Incorrect This would increase gross profit.
d. Incorrect This would increase sales and cost of sales, and the ratio would not change. If
cost of sales is not recorded, gross profit would increase
7.37 a. Incorrect This doesn’t verify that the sales invoices represent actual shipments.
b. Incorrect This would require tracing from shipping documents to invoices.
c. Incorrect This would require tracing from invoices to customer accounts.
d. Correct Vouching is used to establish support for recorded amounts.
7.38 a. Incorrect Unrecorded costs would not increase sales.
b. Incorrect Improper credit approvals would not lower COGS. Goods were shipped for
these sales, and COGS as a percentage of sales would be unchanged.
c. Incorrect Improper sales cutoff would not decrease COGS as a percent of sales.
d. Correct Fictitious sales would increase sales. Because no actual product was shipped,
COGS as a percent of sales would decrease. The most likely debit for fictitious
sales is accounts receivable, causing accounts receivable to increase.
7.39 a. Incorrect Additional inquiries would not provide sufficient corroborating evidence.
b. Correct Reviewing the changes in pricing during the year and ensuring that customers
were charged the new prices provides sufficient, reliable evidence to support the
sales manager’s representation.
c. Incorrect This is an ineffective use of confirmations and requires respondents to identify
unit costs and report information.
d. Incorrect Payments on vendor invoices would not indicate that prices had increased
during the year.
7.40 a. Incorrect When an account is recorded as a receivable, it is already recorded as a revenue.
Adding additional revenue would not cover the theft of accounts receivable.
b. Incorrect Receiving money from petty cash would be a poor method to cover the theft of
accounts receivable. The money in petty cash would have to be accounted for
and is not likely to be sufficient to cover any significant amounts.
c. Incorrect Miscellaneous expense would raise suspicion because all miscellaneous
accounts are high risk and subject to review. In addition, accounts receivable
are usually not written off against an expense.
d. Correct Using the sales returns account would raise the least suspicion because this
account is more commonly linked to accounts receivable. A bookkeeper could
steal money and “write off” to unsuspecting customer’s balance with a fictitious
“sales return.”
7.41 a. Incorrect The payment is probably in transit.
b. Incorrect The shipment is probably in transit.
c. Correct This should have been recorded as a reduction or credit to the receivable by
2/31.
d. Incorrect This occurred after the end of the period.

7-8
7.42 a. Incorrect A schedule of purchases and payments would be used to test transactions and
might be performed.
b. Incorrect Negative confirmations would not be an appropriate choice for large account
balances
c. Incorrect The terms on the accounts receivable would not provide information on balance
and transaction amounts
d. Correct The most likely audit step when there are a few large accounts is to send out
positive confirmations.
7.43 a. Incorrect The aged trial balance provides only indirect evidence about controls.
b. Incorrect The aged trial balance provides no evidence about accuracy.
c. Correct The age of accounts is an indication of credit losses.
d. Incorrect The aged trial balance provides no evidence about existence.
7.44 a. Incorrect Lapping pertains to cash receipts, not sales.
b. Correct False sales journal entries made near the end of the year may have shipping or
other documents that reveal later dates or show lack of sufficient documentation.
c. Incorrect See answer a.
d. Incorrect This step would not detect misappropriation of merchandise.
7.45 a. Incorrect Receiving a confirmation is not evidence that the customer will pay.
b. Incorrect Confirmation will not detect whether the receivables were sold or factored.
c. Correct Accounts receivable confirmation enables recipients to respond that they owe
the company or that they dispute or disagree with the amount the company says
they owe.
d. Incorrect Confirmation provides only indirect evidence that controls are working.
7.46 a Incorrect Prenumbering does not provide any assurance that the document is accutate
b Incorrect Prenumbering does not provide any assurance that the document was recorded
in the proper period.
c. Correct Checking the sequence for missing numbers identifies documents not yet fully
processed in the revenue cycle. It does not provide evidence about accuracy,
cutoff. or occurrence.
d. Incorrect Prenumbering provides no information as to the validity of the transaction.
7.47 a. Correct The accounts receivable debits are supposed to represent sales that have been
ordered by customers and actually shipped to them.
b. Incorrect This is not evidence about existence.
c. Incorrect This provides some evidence about existence, but even if the receivables haven’t
been paid, they may still be valid.
d. Incorrect These file will likely not provide detailed evidence about specific sales.
7.48 a. Incorrect This is an important assertion, but financial statement users are less likely to be
damaged if assets that have not been recorded are found.
b. Correct Financial statement users are more likely to be damaged if assets are found not
to exist or assets are overstated.
c. Incorrect Ownership is important, but doesn’t matter if the assets don’t exist.
d. Incorrect The presentation and disclosure assertion is important but not as important as
existence for asset accounts.

7-9
7.49 c. Correct This is mainly because the other three choices are listed as appropriate work to
do. Also, customers are likely to ignore negative confirmations after earlier
responding to positive confirmations
.
7.50 a. Correct Negative confirmations are most appropriate when the assessed level of risk is
low, dollar balances on accounts are small, and the auditor believes recipients
will give consideration to the confirmations.
b. Incorrect The auditor assumes customers are likely to respond to errors.
c. Incorrect Because negative confirmations offer higher detection risk, risk of material
misstatement should be low when they are used.
d. Incorrect Because negative confirmations offer higher detection risk, risk of material
misstatement should be low when they are used.
7.51 a. Correct Shipments are traced to customers’ invoices. (This does not imply that the
invoices were recorded in the sales journal.)
b. Incorrect See (a) above. The invoice copies need to be traced to the sales journal and
general ledger to determine whether the shipments were recorded as sales.
c. Incorrect Recorded sales were shipped is not established because the sample selection is
from shipments, not from recorded sales.
d. Incorrect See (c) above.
7.52 a. Incorrect Salespeople could write-off accounts for their friends to keep them from having
to pay.
b. Incorrect The credit manager may propose write-offs to reduce days outstanding and
make her or him look better.
c. Correct The treasurer or another high-ranking manager should approve write-offs.
d. Incorrect The cashier could fraudulently collect cash and write off the balance.
7.53 a. Incorrect A second request is the next action that should be performed.
b. Correct Because the confirmations are a sample of the account balance, even immaterial
items should be followed up as they represent other balances in the universe of
receivables.
c. Incorrect Shipping documents should be examined to test the existence of the receivable.
d. Incorrect Client correspondence files may also provide evidence the receivable exists.
7.54 a. Correct Not recording sales on account in the books of original entry is the most
effective way to conceal a subsequent theft of cash receipts. The accounts will
be incomplete but balanced, and procedures applied to the accounting records
will not detect the defalcation.
b. Incorrect The control account wouldn’t match the total of customer accounts.
c. Incorrect Customers would catch the overstatement when examining their statements.
d. Incorrect This is a possibility, but (a) is a better answer. There is less likelihood of getting
caught if the sale is never recorded.
7.55 a. Incorrect The stolen cash wouldn’t be in either of these documents.
b. Incorrect Lapping is not accomplished through write-offs.
c. Correct Lapping is the delayed recording of cash receipts to cover a cash shortage.
Current receipts are posted to the accounts of customers who paid one or two
days previously to avoid complaints (and discovery) when monthly statements
are mailed. The best protection is for the customers to send payments directly to
the company’s depository bank. The next best procedure is to ensure that the
accounts receivable clerk has no access to cash received by the mail room. Thus,
the duties of receiving cash and posting the accounts receivable ledger are
segregated.
d. Incorrect See (a).

7-10
7.56 a. Incorrect A negative confirmation might be used if control risk is low.
b. Correct Because detection risk is lower for positive confirmations than negative
confirmations, a positive confirmation is more likely when inherent risk is high.
c. Incorrect Whether the account is due or not usually doesn’t affect the type of
confirmation. However, if it is long past due, a positive confirmation is more
appropriate.
d. Incorrect A related-party account could be a factor that influences a decision to send a
positive confirmation. The fact that this account was not a related party would
likely lead the auditor to choose a negative confirmation.
7.57 a. Incorrect Since a large portion of the sales occur in the last month of the year using
analytical procedures at an interim date would not be effective.
b. Correct Since a large portion of the sales occur in the last month the auditor needs to test
end of year sales, specifically the determination of the timing of sales is
important to ensure sales were recorded in the proper period.
C. Incorrect Since a large portion of the sales occur in the last month of the year using testing
internal controls at an interim date would not be effective in determining that
year-end sales were accurate. Additional testing through year end would be
required.
D. Incorrect Period-end compensation may or may not be based on sales. Even if period-end
compensation is tied to sales, a review of the compensation would not provide
evidence of the valuation or cut-off of sales.
7.58 a. Incorrect There is no requirement to confirm accounts receivable that proceed the prior
year.
b. Correct Confirmation are generally reserved for accounts that are material to the balance
being examined, in this case, accounts receivable.
C. Incorrect There is no reason for management to intentionally understate accounts
receivable. Therefore, this would not be a required account for confirmation.
D. Incorrect If an account is subject to valuation estimates the auditor needs to review the
underlying assumptions of those estimates. Confirming the account will not
provide any information regarding the validity of the estimate.
7.59 a. Incorrect Replacement of an accounts receivable with another confirmation is not an
acceptable procedure in any situation. The auditor should try to verify that the
fax received actually came from an appropriate person at the client. If the
auditor cannot verify the legitimacy of the confirmation, the confirmation should
be classified as an exception and detailed testing of the underlying transactions
should occur.
b. Incorrect If the auditor can verify the legitimacy of the confirmation, the confirmation
may be accepted. Therefore, before classifying the accounts receivable as an
exception the auditor should attempt to verify the source. Consequently, answer
D is the better answer.
C. Incorrect Accepting the accounts receivable without verifying that the fax received
actually came from the appropriate person at the client is not an acceptable
procedure.
D. Correct When a reply to a confirmation is received via fax the auditor must determine
that the fax actually came from the appropriate person at the client. A telephone
call to an appropriate person at the audit client is an acceptable method for
verifying the legitimacy of the response.

7-11
SOLUTIONS FOR EXERCISES, PROBLEMS, AND
SIMULATIONS
7.60 Control Objectives and Procedures Associations
a. “Occurrence” Sales recorded, goods not shipped
b. “Completeness” Goods shipped, sales not recorded
c. “Accuracy” Goods shipped to a bad credit risk customer
d. “Accuracy” Sales billed at the wrong price or wrong quantity
e. “Classification” Product line A sales recorded as Product line B
f. “Completeness” Failure to post charges to customers for sales
g. “Cutoff” January sales recorded in December
CONTROL PROCEDURES
1. Sales order approved for credit X
2. Prenumbered shipping doc prepared, sequence checked X X
3. Shipping document quantity compared to sales invoice X X X
4. Prenumbered sales invoices, sequence checked X
5. Sales invoice checked to sales order X
6. Invoiced prices compared to approved price list X
7. General ledger code checked for sales product lines X
8. Sales dollar batch totals compared to sales journal X X X
9. Periodic sales total compared to same period accounts
receivable postings
X
10. Accountants have instructions to date sales on the date of
shipment
X
11. Sales entry date compared to shipping doc date X
12. Accounts receivable subsidiary totaled and reconciled to
accounts receivable control account
X
13. Intercompany accounts reconciled with subsidiary company
records
X
14. Credit files updated for customer payment history X
15. Overdue customer accounts investigated for collection X X X X

7-12
7.60 Control Objectives and Procedures Associations (Continued)
EXHIBIT 7.57-1 Blank Form for Students
a. Sales recorded, goods not shipped
b. Goods shipped, sales not recorded
c. Goods shipped to a bad credit risk customer
d. Sales billed at the wrong price or wrong quantity
e. Product line A sales recorded as Product line B
f. Failure to post charges to customers for sales
g. January sales recorded in December
CONTROL PROCEDURES
1. Sales order approved for credit
2. Prenumbered shipping doc prepared. sequence checked
3. Shipping document quantity compared to sales invoice
4. Prenumbered sales invoices, sequence checked
5. Sales invoice checked to sales order
6. Invoiced prices compared to approved price list
7. General ledger code checked for sales product lines
8. Sales dollar batch totals compared to sales journal
9. Periodic sales total compared to same period accounts
receivable postings
10. Accountants have instructions to date sales on the date of
shipment
11. Sales entry date compared to shipping doc date
12. Accounts receivable subsidiary totaled and reconciled to
accounts receivable control account
13. Intercompany accounts reconciled with subsidiary company
records
14. Credit files updated for customer payment history
15. Overdue customer accounts investigated for collection
7.61 Control Assertion Associations
Error Assertions
a. Sales recorded,
goods not shipped
Occurrence
b. Goods shipped,
sales not recorded
Completeness
c. Goods shipped to
a bad credit risk
customer
Accuracy
d. Sales billed at the
wrong price or
wrong quantity
Accuracy
e. Product A sales
recorded as
Product line B
Classification

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Galileo’s discovery concerning the composition of forces, D’Alembert
“established, for the first time, the equations of equilibrium of any
system of forces applied to the different points of a solid body”—
equations which include all cases of levers and an infinity of cases
besides. Clearly this is progress towards a higher generality—
towards a knowledge more independent of special circumstances—
towards a study of phenomena “the most disengaged from the
incidents of particular cases;” which is M. Comte’s definition of “the
most simple phenomena.” Does it not indeed follow from the
admitted fact, that mental advance is from the concrete to the
abstract, from the particular to the general, that the universal and
therefore most simple truths are the last to be discovered? Should
we ever succeed in reducing all orders of phenomena to some single
law—say of atomic action, as M. Comte suggests—must not that law
answer to his test of being independent of all others, and therefore
most simple? And would not such a law generalize the phenomena
of gravity, cohesion, atomic affinity, and electric repulsion, just as
the laws of number generalize the quantitative phenomena of space,
time and force?
The possibility of saying so much in support of an hypothesis the
very reverse of M. Comte’s, at once proves that his generalization is
only a half-truth. The fact is that neither proposition is correct by
itself; and the actuality is expressed only by putting the two
together. The progress of science is duplex. It is at once from the
special to the general, and from the general to the special. It is
analytical and synthetical at the same time.
M. Comte himself observes that the evolution of science {25} has
been accomplished by the division of labour; but he quite misstates
the mode in which this division of labour has operated. As he
describes it, it has been simply an arrangement of phenomena into
classes, and the study of each class by itself. He does not recognize
the effect of progress in each class upon all other classes: he
recognizes only the effect on the class succeeding it in his

hierarchical scale. Or if he occasionally admits collateral influences
and inter
com
mun
i
ca
tions, he does it so grudgingly, and so quickly
puts the admissions out of sight and forgets them, as to leave the
impression that, with but trifling exceptions, the sciences aid one
another only in the order of their alleged succession. The fact is,
however, that the division of labour in science, like the division of
labour in society, and like the “physiological division of labour” in
individual organisms, has been not only a specialization of functions,
but a continuous helping of each division by all the others, and of all
by each. Every particular class of inquirers has, as it were, secreted
its own particular order of truths from the general mass of material
which observation accumulates; and all other classes of inquirers
have made use of these truths as fast as they were elaborated, with
the effect of enabling them the better to elaborate each its own
order of truths. It was thus in sundry of the cases we have quoted
as at variance with M. Comte’s doctrine. It was thus with the
application of Huyghens’s optical discovery to astronomical
observation by Galileo. It was thus with the application of the
isochronism of the pendulum to the making of instruments for
measuring intervals, astronomical and other. It was thus when the
discovery that the refraction and dispersion of light did not follow the
same law of variation, affected both astronomy and physiology by
giving us achromatic telescopes and microscopes. It was thus when
Bradley’s discovery of the aberration of light enabled him to make
the first step towards ascertaining the motions of the stars. {26} It
was thus when Cavendish’s torsion-balance experiment determined
the specific gravity of the Earth, and so gave a datum for calculating
the specific gravities of the Sun and Planets. It was thus when tables
of atmospheric refraction enabled observers to write down the real
places of the heavenly bodies instead of their apparent places. It
was thus when the discovery of the different expansibilities of metals
by heat, gave us the means of correcting our chronometrical
measurements of astronomical periods. It was thus when the lines of

the prismatic spectrum were used to distinguish the heavenly bodies
that are of like nature with the sun from those which are not. It was
thus when, as recently, an elec
tro-tel
e
graphic instrument was
invented for the more accurate registration of meridional transits. It
was thus when the difference in the rates of a clock at the equator,
and nearer the poles, gave data for calculating the oblateness of the
earth, and accounting for the precession of the equinoxes. It was
thus—but it is needless to continue. Here, within our own limited
knowledge of its history, we have named ten additional cases in
which the single science of astronomy has owed its advance to
sciences coming after it in M. Comte’s series. Not only its minor
changes, but its greatest revolutions have been thus determined.
Kepler could not have discovered his celebrated laws had it not been
for Tycho Brahe’s accurate observations; and it was only after some
progress in physical and chemical science that the improved
instruments with which those observations were made, became
possible. The heliocentric theory of the Solar System had to wait
until the invention of the telescope before it could be finally
established. Nay, even the grand discovery of all—the law of
gravitation—depended for its proof upon an operation of physical
science, the measurement of a degree on the Earth’s surface. So
completely, indeed, did it thus depend, that Newton had actually
abandoned his hypothesis because the {27} length of a degree, as
then stated, brought out wrong results; and it was only after Picart’s
more exact measurement was published, that he returned to his
calculations and proved his great generalization. Now this constant
intercommunion which, for brevity’s sake, we have illustrated in the
case of one science only, has been taking place with all the sciences.
Throughout the whole course of their evolution there has been a
continuous consensus of the sciences—a consensus exhibiting a
general correspondence with the consensus of the faculties in each
phase of mental development; the one being an objective registry of
the subjective state of the other.

From our present point of view, then, it becomes obvious that the
conception of a serial arrangement of the sciences is a vicious one.
It is not simply that, as M. Comte admits, such a clas
si
fi
ca
tion “will
always involve something, if not arbitrary, at least artificial;” it is not,
as he would have us believe, that, neglecting minor imperfections
such a clas
si
fi
ca
tion may be substantially true; but it is that any
grouping of the sciences in a succession gives a radically erroneous
idea of their genesis and their dependencies. There is no “one
rational order among a host of possible systems.” There is no “true
filiation of the sciences.” The whole hypothesis is fundamentally
false. Indeed, it needs but a glance at its origin to see at once how
baseless it is. Why a series? What reason have we to suppose that
the sciences admit of a linear arrangement? Where is our warrant
for assuming that there is some succession in which they can be
placed? There is no reason; no warrant. Whence then has arisen the
supposition? To use M. Comte’s own phraseology, we should say, it is
a metaphysical conception. It adds another to the cases constantly
occurring, of the human mind being made the measure of Nature.
We are obliged to think in sequence; it is a law of our minds that we
must consider subjects separately, one after another: therefore {28}
Nature must be serial—therefore the sciences must be classifiable in
a succession. See here the birth of the notion, and the sole evidence
of its truth. Men have been obliged when arranging in books their
schemes of education and systems of knowledge, to choose some
order or other. And from inquiring what is the best order, have fallen
into the belief that there is an order which truly represents the facts
—have persevered in seeking such an order; quite overlooking the
previous question whether it is likely that Nature has consulted the
convenience of book-making. For German philosophers, who hold
that Nature is “petrified intelligence,” and that logical forms are the
foundations of all things, it is a consistent hypothesis that as thought
is serial, Nature is serial; but that M. Comte, who is so bitter an

opponent of all an
thro
po
mor
phism, even in its most evanescent
shapes, should have committed the mistake of imposing upon the
external world an arrangement which so obviously springs from a
limitation of the human con
scious
ness, is somewhat strange. And it
is the more strange when we call to mind how, at the outset, M.
Comte remarks that in the beginning “toutes les sciences sont
cultivées simultanément par les mêmes esprits;” that this is
“inevitable et même indispensable;” and how he further remarks
that the different sciences are “comme les diverses branches d’un
tronc unique.” Were it not accounted for by the distorting influence
of a cherished hypothesis, it would be scarcely possible to
understand how, after recognizing truths like these, M. Comte should
have persisted in attempting to construct “une échelle
encyclopédique.”
The metaphor which M. Comte has here so inconsistently used to
express the relations of the sciences—branches of one trunk—is an
approximation to the truth, though not the truth itself. It suggests
the facts that the sciences had a common origin; that they have
been developing simultaneously; and that they have been from time
to time dividing and sub-dividing. But it fails to suggest the fact, that
the {29} divisions and sub-divisions thus arising do not remain
separate, but now and again re-unite in direct and indirect ways.
They inosculate; they severally send off and receive connecting
growths; and the intercommunion has been ever becoming more
frequent, more intricate, more widely ramified. There has all along
been higher specialization, that there might be a larger
generalization; and a deeper analysis, that there might be a better
synthesis. Each larger generalization has lifted sundry specializations
still higher; and each better synthesis has prepared the way for still
deeper analysis.
And here we may fitly enter upon the task awhile since indicated—
a sketch of the Genesis of Science, regarded as a gradual outgrowth
from common knowledge—an extension of the perceptions by the

aid of the reason. We propose to treat it as a psychological process
historically displayed; tracing at the same time the advance from
qualitative to quantitative prevision; the progress from concrete facts
to abstract facts, and the application of such abstract facts to the
analysis of new orders of concrete facts; the simultaneous advance
in generalization and specialization; the continually increasing
subdivision and reunion of the sciences; and their constantly
improving consensus.
To trace out scientific evolution from its deepest roots would, of
course, involve a complete analysis of the mind. For as science is a
development of that common knowledge acquired by the unaided
senses and uncultured reason, so is that common knowledge itself
gradually built up out of the simplest perceptions. We must,
therefore, begin somewhere abruptly; and the most appropriate
stage to take for our point of departure will be the adult mind of the
savage.
Commencing thus, without a proper preliminary analysis, we are
naturally somewhat at a loss how to present, in a satisfactory
manner, those fundamental processes of thought out of which
science originates. Perhaps our argument may {30} be best initiated
by the proposition, that all intelligent action whatever depends upon
the discerning of distinctions among surrounding things. The
condition under which only it is possible for any creature to obtain
food and avoid danger, is, that it shall be differently affected by
different objects—that it shall be led to act in one way by one
object, and in another way by another. In the lower orders of
creatures this condition is fulfilled by means of an apparatus which
acts automatically. In the higher orders the actions are partly
automatic, partly conscious. And in man they are almost wholly
conscious. Throughout, however, there must necessarily exist a
certain clas
si
fi
ca
tion of things according to their properties—a clas
si
‐

fi
ca
tion which is either organically registered in the system, as in the
inferior creation, or is formed by conscious experience, as in
ourselves. And it may be further remarked, that the extent to which
this clas
si
fi
ca
tion is carried, roughly indicates the height of
intelligence—that, while the lowest organisms are able to do little
more than discriminate organic from inorganic matter; while the
generality of animals carry their clas
si
fi
ca
tions no further than to a
limited number of plants or creatures serving for food, a limited
number of beasts of prey, and a limited number of places and
materials; the most degraded of the human race possess a
knowledge of the distinctive natures of a great variety of substances,
plants, animals, tools, persons, &c.; not only as classes but as
individuals.
What now is the mental process by which clas
si
fi
ca
tion is
effected? Manifestly it is a recognition of the likeness or unlikeness
of things, either in respect of their sizes, colours, forms, weights,
textures, tastes, &c., or in respect of their modes of action. By some
special mark, sound, or motion, the savage identifies a certain four-
legged creature he sees, as one that is good for food, and to be
caught in a particular way; or as one that is dangerous; and acts
accordingly. He has classed together all the creatures that are alike
in {31} this particular. And manifestly in choosing the wood out of
which to form his bow, the plant with which to poison his arrows,
the bone from which to make his fish-hooks, he identifies them
through their chief sensible properties as belonging to the general
classes, wood, plant, and bone, but distinguishes them as belonging
to sub-classes by virtue of certain properties in which they are
unlike the rest of the general classes they belong to; and so forms
genera and species.
And here it becomes manifest that not only is clas
si
fi
ca
tion carried
on by grouping together in the mind things that are like; but that
classes and sub-classes are formed and arranged according to the
degrees of unlikeness. Things strongly contrasted are alone

distinguished in the lower stages of mental evolution; as may be any
day observed in an infant. And gradually as the powers of
discrimination increase, the strong
ly-con
trast
ed classes at first
distinguished, come to be each divided into sub-classes, differing
from each other less than the classes differ; and these sub-classes
are again divided after the same manner. By the continuance of
which process, things are gradually arranged into groups, the
members of which are less and less unlike; ending, finally, in groups
whose members differ only as individuals, and not specifically. And
thus there tends ultimately to arise the notion of complete likeness.
For manifestly, it is impossible that groups should continue to be
subdivided in virtue of smaller and smaller differences, without there
being a simultaneous approximation to the notion of no difference.
Let us next notice that the recognition of likeness and unlikeness,
which underlies clas
si
fi
ca
tion, and out of which continued clas
si
fi
ca
‐
tion evolves the idea of complete likeness—let us next notice that it
also underlies the process of naming, and by consequence
language. For all language consists, at the outset, of symbols which
are as like to the things symbolized as it is practicable to make
them. The {32} language of signs is a means of conveying ideas by
mimicking the actions or peculiarities of the things referred to.
Verbal language also, in its first stage, is a mode of suggesting
objects or acts by imitating the sounds which the objects make, or
with which the acts are accompanied. Originally these two languages
were used simultaneously. It needs but to watch the gesticulations
with which the savage accompanies his speech—to see a Bushman
dramatizing before an audience his mode of catching game—or to
note the extreme paucity of words in primitive vocabularies; to infer
that in the beginning, attitudes, gestures, and sounds, were all
combined to produce as good a likeness as possible of the things,
animals, persons, or events described; and that as the sounds came
to be understood by themselves the gestures fell into disuse: leaving
traces, however, in the manners of the more excitable civilized races.

But be this as it may, it suffices simply to observe, how many of the
words current among barbarous peoples are like the sounds
appertaining to the things signified; how many of our own oldest
and simplest words have the same peculiarity; how children
habitually invent imitative words; and how the sign-language
spontaneously formed by deaf mutes is based on imitative actions—
to be convinced that the notion of likeness is that from which the
nomenclature of objects takes its rise. Were there space we might
go on to point out how this law of likeness is traceable, not only in
the origin but in the development of language; how in primitive
tongues the plural is made by a duplication of the singular, which is
a multiplication of the word to make it like the multiplicity of the
things; how the use of metaphor—that prolific source of new words
—is a suggesting of ideas which are like the ideas to be conveyed in
some respect or other; and how, in the copious use of simile, fable,
and allegory among uncivilized races, we see that complex
conceptions which there is no direct language for, are {33} rendered,
by presenting known conceptions more or less like them.
This view is confirmed, and the predominance of this notion of
likeness in primitive thought further illustrated, by the fact that our
system of presenting ideas to the eye originated after the same
fashion. Writing and printing have descended from picture-language.
The earliest mode of permanently registering a fact was by depicting
it on a skin and afterwards on a wall; that is—by exhibiting
something as like to the thing to be remembered as it could be
made. Gradually as the practice grew habitual and extensive, the
most frequently repeated forms became fixed, and presently
abbreviated; and, passing through the hieroglyphic and ideographic
phases, the symbols lost all apparent relation to the things signified:
just as the majority of our spoken words have done.
Observe, again, that the same thing is true respecting the genesis
of reasoning. The likeness which is perceived to exist between
cases, is the essence of all early reasoning and of much of our

present reasoning. The savage, having by experience discovered a
relation between a certain object and a certain act, infers that the
like relation will be found in future. And the expressions we use in
our arguments—“analogy implies,” “the cases are not parallel,” “by
parity of reasoning,” “there is no similarity,”—show how constantly
the idea of likeness underlies our ratiocinative processes. Still more
clearly will this be seen on recognizing the fact that there is a close
connexion between reasoning and clas
si
fi
ca
tion; that the two have a
common root; and that neither can go on without the other. For on
the one hand, it is a familiar truth that the attributing to a body in
consequence of some of its properties, all those other properties in
virtue of which it is referred to a particular class, is an act of
inference. And, on the other hand, the forming of a generalization is
the putting together in one class, all those {34} cases which present
like relations; while the drawing a deduction is essentially the
perception that a particular case belongs to a certain class of cases
previously generalized. So that as clas
si
fi
ca
tion is a grouping
together of like things; reasoning is a grouping together of like
relations among things. Add to which, that while the perfection
gradually achieved in clas
si
fi
ca
tion consists in the formation of
groups of objects which are completely alike; the perfection
gradually achieved in reasoning consists in the formation of groups
of cases which are completely alike.
Once more we may contemplate this dominant idea of likeness as
exhibited in art. All art, civilized as well as savage, consists almost
wholly in the making of objects like other objects; either as found in
Nature, or as produced by previous art. If we trace back the varied
art-products now existing, we find that at each stage the divergence
from previous patterns is but small when compared with the
agreement; and in the earliest art the persistency of imitation is yet
more conspicuous. The old forms and ornaments and symbols were
held sacred, and perpetually copied. Indeed, the strong imitative
tendency notoriously displayed by the lowest human races—often

seeming to be half automatic, ensures among them a constant
reproducing of likenesses of things, forms, signs, sounds, actions
and whatever else is imitable; and we may even suspect that this
aboriginal peculiarity is in some way connected with the culture and
development of this general conception, which we have found so
deep and wide-spread in its applications.
And now let us go on to consider how, by a further unfolding of
this same fundamental notion, there is a gradual formation of the
first germs of science. This idea of likeness which underlies clas
si
fi
‐
ca
tion, nomenclature, language spoken and written, reasoning, and
art; and which plays so important a part because all acts of
intelligence are made {35} possible only by distinguishing among
surrounding things, or grouping them into like and unlike;—this idea
we shall find to be the one of which science is the especial product.
Already during the stage we have been describing, there has existed
qualitative prevision in respect to the commoner phenomena with
which savage life is familiar; and we have now to inquire how the
elements of quantitative prevision are evolved. We shall find that
they originate by the perfecting of this same idea of likeness—that
they have their rise in that conception of complete likeness which,
as we have seen, necessarily results from the continued process of
clas
si
fi
ca
tion.
For when the process of clas
si
fi
ca
tion has been carried as far as it
is possible for the uncivilized to carry it—when the animal kingdom
has been grouped not merely into quadrupeds, birds, fishes, and
insects, but each of these divided into kinds—when there come to be
classes, in each of which the members differ only as individuals, and
not specifically; it is clear that there must frequently occur an
observation of objects which differ so little as to be in
dis
tin
guish
able.
Among several creatures which the savage has killed and carried
home, it must often happen that some one, which he wished to
identify, is so exactly like another that he cannot tell which is which.
Thus, then, there originates the notion of equality. The things which

among ourselves are called equal—whether lines, angles, weights,
temperatures, sounds or colours—are things which produce in us
sensations which cannot be distinguished from each other. It is true
that we now apply the word equal chiefly to the separate traits or
relations which objects exhibit, and not to those combinations of
them constituting our conceptions of the objects; but this limitation
of the idea has evidently arisen by analysis. That the notion of
equality originated as alleged, will, we think, become obvious on
remembering that as there were no artificial objects from which it
could have been {36} abstracted, it must have been abstracted from
natural objects; and that the various families of the animal kingdom
chiefly furnish those natural objects which display the requisite
exactitude of likeness.
The experiences out of which this general idea of equality is
evolved, give birth at the same time to a more complex idea of
equality; or, rather, the process just described generates an idea of
equality which further experience separates into two ideas—equality
of things and equality of relations. While organic forms occasionally
exhibit this perfection of likeness out of which the notion of simple
equality arises, they more frequently exhibit only that kind of
likeness which we call similarity; and which is really compound
equality. For the similarity of two creatures of the same species but
of different sizes, is of the same nature as the similarity of two
geometrical figures. In either case, any two parts of the one bear
the same ratio to one another, as the homologous parts of the other.
Given in a species, the proportions found to exist among the bones,
and we may, and zoologists do, predict from any one, the
dimensions of the rest; just as, when knowing the proportions
subsisting among the parts of a geometrical figure, we may, from
the length of one, calculate the others. And if, in the case of similar
geometrical figures, the similarity can be established only by proving
exactness of proportion among the homologous parts—if we express
this relation between two parts in the one, and the corresponding

parts in the other, by the formula A is to B as a is to b; if we
otherwise write this, A to B = a to b; if, consequently, the fact we
prove is that the relation of A to B equals the relation of a to b;
then it is manifest that the fundamental conception of similarity is
equality of relations. With this explanation we shall be understood
when we say that the notion of equality of relations is the basis of all
exact reasoning. Already it has been shown that reasoning in
general is a recognition {37} of likeness of relations; and here we
further find that while the notion of likeness of things ultimately
evolves the idea of simple equality, the notion of likeness of relations
evolves the idea of equality of relations: of which the one is the
concrete germ of exact science, while the other is its abstract germ.
Those who cannot understand how the recognition of similarity in
creatures of the same kind, can have any alliance with reasoning,
will get over the difficulty on remembering that the phenomena
among which equality of relations is thus perceived, are phenomena
of the same order and are present to the senses at the same time;
while those among which developed reason perceives relations, are
generally neither of the same order, nor simultaneously present. And
if, further, they will call to mind how Cuvier and Owen, from a single
part of a creature, as a tooth, construct the rest by a process of
reasoning based on this equality of relations, they will see that the
two things are intimately connected, remote as they at first seem.
But we anticipate. What it concerns us here to observe is, that from
familiarity with organic forms there simultaneously arose the ideas of
simple equality, and equality of relations.
At the same time, too, and out of the same mental processes,
came the first distinct ideas of number. In the earliest stages, the
presentation of several like objects produced merely an indefinite
conception of multiplicity; as it still does among Australians, and
Bushmen, and Damaras, when the number presented exceeds three
or four. With such a fact before us we may safely infer that the first
clear numerical conception was that of duality as contrasted with

unity. And this notion of duality must necessarily have grown up side
by side with those of likeness and equality; seeing that it is
impossible to recognize the likeness of two things without also
perceiving that there are two. From the very beginning the
conception of number must have been, as it is still, associated with
{38} likeness or equality of the things numbered; and for the
purposes of calculation, an ideal equality of the things is assumed.
Before any absolutely true numerical results can be reached, it is
requisite that the units be absolutely equal. The only way in which
we can establish a numerical relationship between things that do not
yield us like impressions, is to divide them into parts that do yield us
like impressions. Two unlike magnitudes of extension, force, time,
weight, or what not, can have their relative amounts estimated, only
by means of some small unit that is contained many times in both;
and even if we finally write down the greater one as a unit and the
other as a fraction of it, we state, in the denominator of the fraction,
the number of parts into which the unit must be divided to be
comparable with the fraction. It is, indeed, true, that by a modern
process of abstraction, we occasionally apply numbers to unequal
units, as the furniture at a sale or the various animals on a farm,
simply as so many separate entities; but no exact quantitative result
can be brought out by calculation with units of this order. And,
indeed, it is the distinctive peculiarity of the calculus in general, that
it proceeds on the hypothesis of that absolute equality of its abstract
units, which no real units possess; and that the exactness of its
results holds only in virtue of this hypothesis. The first ideas of
number must necessarily then have been derived from like or equal
magnitudes as seen chiefly in organic objects; and as the like
magnitudes most frequently observed were magnitudes of
extension, it follows that geometry and arithmetic had a
simultaneous origin.
Not only are the first distinct ideas of number co-ordinate with
ideas of likeness and equality, but the first efforts at numeration

display the same relationship. On reading accounts of savage tribes,
we find that the method of counting by the fingers, still followed by
many children, is the aboriginal method. Neglecting the several
cases {39} in which the ability to enumerate does not reach even to
the number of fingers on one hand, there are many cases in which it
does not extend beyond ten—the limit of the simple finger notation.
The fact that in so many instances, remote, and seemingly unrelated
nations, have adopted ten as their basic number; together with the
fact that in the remaining instances the basic number is either five
(the fingers of one hand) or twenty (the fingers and toes); of
themselves show that the fingers were the original units of
numeration. The still surviving use of the word digit, as the general
name for a figure in arithmetic, is significant; and it is even said that
our word ten (Sax. tyn; Dutch, tien; German, zehn) means in its
primitive expanded form two hands. So that, originally, to say there
were ten things, was to say there were two hands of them. From all
which evidence it is tolerably clear that the earliest mode of
conveying the idea of a number of things, was by holding up as
many fingers as there were things; that is, by using a symbol which
was equal, in respect of multiplicity, to the group symbolized. For
which inference there is, indeed, strong confirmation in the
statement that our own soldiers spontaneously adopted this device
in their dealings with the Turks during the Crimean war. And here it
should be remarked that in this re-combination of the notion of
equality with that of multiplicity, by which the first steps in
numeration are effected, we may see one of the earliest of those
inosculations between the diverging branches of science, which are
afterwards of perpetual occurrence.
As this observation suggests, it will be well, before tracing the
mode in which exact science emerges from the inexact judgments of
the senses, and showing the non-serial evolution of its divisions, to
note the non-serial character of those preliminary processes of which
all after development is a continuation. On re-considering them it

will be seen that not only are they divergent branches {40} from a
common root,—not only are they simultaneous in their growth; but
that they are mutual aids; and that none can advance without the
rest. That progress of clas
si
fi
ca
tion for which the unfolding of the
perceptions paves the way, is impossible without a corresponding
progress in language, by which greater varieties of objects are
thinkable and expressible. On the one hand clas
si
fi
ca
tion cannot be
carried far without names by which to designate the classes; and on
the other hand language cannot be made faster than things are
classified. Again, the multiplication of classes and the consequent
narrowing of each class, itself involves a greater likeness among the
things classed together; and the consequent approach towards the
notion of complete likeness itself allows clas
si
fi
ca
tion to be carried
higher. Moreover, clas
si
fi
ca
tion necessarily advances pari passu with
rationality—the clas
si
fi
ca
tion of things with the clas
si
fi
ca
tion of
relations. For things that belong to the same class are, by
implication, things of which the properties and modes of behaviour—
the co-existences and sequences—are more or less the same; and
the recognition of this sameness of co-existences and sequences is
reasoning. Whence it follows that the advance of clas
si
fi
ca
tion is
necessarily proportionate to the advance of gen
er
al
i
za
tions. Yet
further, the notion of likeness, both in things and relations,
simultaneously evolves by one process of culture the ideas of
equality of things and equality of relations; which are the respective
bases of exact concrete reasoning and exact abstract reasoning—
Mathematics and Logic. And once more, this idea of equality, in the
very process of being formed, necessarily gives origin to two series
of relations—those of magnitude and those of number; from which
arise geometry and the calculus. Thus the process throughout is one
of perpetual subdivision and perpetual inter
com
mun
i
ca
tion of the
divisions. From the very first there has been that consensus of
different kinds of knowledge, {41} answering to the consensus of the

intellectual faculties, which, as already said, must exist among the
sciences.
Let us now go on to observe how, out of the notions of equality
and number, as arrived at in the manner described, there gradually
arose the elements of quantitative prevision.
Equality, once having come to be definitely conceived, was
recognizable among other phenomena than those of magnitude.
Being predicable of all things producing in
dis
tin
guish
able
impressions, there naturally grew up ideas of equality in weights,
sounds, colours, &c.; and, indeed, it can scarcely be doubted that
the occasional experience of equal weights, sounds, and colours,
had a share in developing the abstract conception of equality—that
the ideas of equality in sizes, relations, forces, resistances, and
sensible properties in general, were evolved during the same stage
of mental development. But however this may be, it is clear that as
fast as the notion of equality gained definiteness, so fast did that
lowest kind of quantitative prevision which is achieved without any
instrumental aid, become possible. The ability to estimate, however
roughly, the amount of a foreseen result, implies the conception that
it will be equal to a certain imagined quantity; and the correctness
of the estimate will manifestly depend on the precision which the
perceptions of sensible equality have reached. A savage with a piece
of stone in his hand, and another piece lying before him of greater
bulk but of the same kind (sameness of kind being inferred from the
equality of the two in colour and texture) knows about what effort
he must put forth to raise this other piece; and he judges accurately
in proportion to the accuracy with which he perceives that the one is
twice, three times, four times, &c. as large as the other; that is—in
proportion to the precision of his ideas of equality and number. And
here let us not omit to notice that even in these vaguest of
quantitative previsions, the conception of equality of relations is also
involved. For it is only in {42} virtue of an undefined con
scious
ness
that the relation between bulk and weight in the one stone is equal

to the relation between bulk and weight in the other, that even the
roughest approximation can be made.
But how came the transition from those uncertain perceptions of
equality which the unaided senses give, to the certain ones with
which science deals? It came by placing the things compared in
juxtaposition. Equality being asserted of things which give us in
dis
‐
tin
guish
able impressions, and no distinct comparison of impressions
being possible unless they occur in immediate succession, it results
that exactness of equality is ascertainable in proportion to the
closeness of the compared things. Hence the fact that when we wish
to judge of two shades of colour whether they are alike or not, we
place them side by side; hence the fact that we cannot, with any
precision, say which of two allied sounds is the louder, or the higher
in pitch, unless we hear the one immediately after the other; hence
the fact that to estimate the ratio of weights, we take one in each
hand, that we may compare their pressures by rapidly alternating in
thought from the one to the other; hence the fact, that in a piece of
music, we can continue to make equal beats when the first beat has
been given, but cannot ensure commencing with the same length of
beat on a future occasion; and hence, lastly, the fact, that of all
magnitudes, those of linear extension are those of which the
equality is most precisely ascertainable, and those to which, by
consequence, all others have to be reduced. For it is the peculiarity
of linear extension that it alone allows its magnitudes to be placed in
absolute juxtaposition, or, rather, in coincident position; it alone can
test the equality of two magnitudes by observing whether they will
coalesce, as two equal mathematical lines do, when placed between
the same points; it alone can test equality by trying whether it will
become identity. Hence, then, the fact, that all exact science is
reducible, {43} by an ultimate analysis, to results measured in equal
units of linear extension.
Still it remains to be noticed in what manner this determination of
equality by comparison of linear magnitudes originated. Once more

may we perceive that surrounding natural objects supplied the
needful lessons. From the beginning there must have been a
constant experience of like things placed side by side—men standing
and walking together; animals from the same herd; fish from the
same shoal. And the ceaseless repetition of these experiences could
not fail to suggest the observation, that the nearer together any
objects were, the more visible became any inequality between them.
Hence the obvious device of putting in apposition, things of which it
was desired to ascertain the relative magnitudes. Hence the idea of
measure. And here we suddenly come upon a group of facts which
afford a solid basis to the remainder of our argument; while they
also furnish strong evidence in support of the foregoing speculations.
Those who look sceptically on this attempted rehabilitation of early
mental development, and who think that the derivation of so many
primary notions from organic forms is somewhat strained, will
perhaps see more probability in the hypotheses which have been
ventured, on discovering that all measures of extension and force
originated from the lengths and weights of organic bodies, and all
measures of time from the periodic phenomena of either organic or
inorganic bodies.
Thus, among linear measures, the cubit of the Hebrews was the
length of the forearm from the elbow to the end of the middle
finger; and the smaller scriptural dimensions are expressed in hand-
breadths and spans. The Egyptian cubit, which was similarly
derived, was divided into digits, which were finger-breadths; and
each finger-breadth was more definitely expressed as being equal to
four grains of barley placed breadthwise. Other ancient measures
were {44} the orgyia or stretch of the arms, the pace, and the palm.
So persistent has been the use of these natural units of length in the
East, that even now some Arabs mete out cloth by the forearm. So,
too, is it with European measures. The foot prevails as a dimension
throughout Europe, and has done so since the time of the Romans,
by whom, also, it was used: its lengths in different places varying

not much more than men’s feet vary. The heights of horses are still
expressed in hands. The inch is the length of the terminal joint of
the thumb; as is clearly shown in France, where pouce means both
thumb and inch. Then we have the inch divided into three barley-
corns. So completely, indeed, have these organic dimensions served
as the substrata of mensuration, that it is only by means of them
that we can form any estimate of some of the ancient distances. For
example, the length of a degree on the Earth’s surface, as
determined by the Arabian astronomers shortly after the death of
Haroun-al-Raschid, was fifty-six of their miles. We know nothing of
their mile further than that it was 4000 cubits; and whether these
were sacred cubits or common cubits, would remain doubtful, but
that the length of the cubit is given as twenty-seven inches, and
each inch defined as the thickness of six barley-grains. Thus one of
the earliest measurements of a degree comes down to us in barley-
grains. Not only did organic lengths furnish those approximate
measures which satisfied men’s needs in ruder ages, but they
furnished also the standard measures required in later times. One
instance occurs in our own history. To remedy the irregularities then
prevailing, Henry I. commanded that the ulna, or ancient ell, which
answers to the modern yard, should be made of the exact length of
his own arm.
Measures of weight had a kindred derivation. Seeds seem
commonly to have supplied the units. The original of the carat used
for weighing in India is a small bean. Our own systems, both troy
and avoirdupois, are derived {45} primarily from wheat-corns. Our
smallest weight, the grain is a grain of wheat. This is not a
speculation; it is an his
tor
i
cal
ly-reg
is
tered fact. Henry III. enacted
that an ounce should be the weight of 640 dry grains of wheat from
the middle of the ear. And as all the other weights are multiples or
sub-multiples of this, it follows that the grain of wheat is the basis of
our scale. So natural is it to use organic bodies as weights, before
artificial weights have been established, or where they are not to be

had, that in some of the remoter parts of Ireland the people are said
to be in the habit, even now, of putting a man into the scales to
serve as a measure for heavy commodities.
Similarly with time. Astronomical periodicity, and the periodicity of
animal and vegetable life, are simultaneously used in the first stages
of progress for estimating epochs. The simplest unit of time, the day,
nature supplies ready made. The next simplest period, the moneth
or month, is also thrust upon men’s notice by the conspicuous
changes constituting a lunation. For larger divisions than these, the
phenomena of the seasons, and the chief events from time to time
occurring, have been used by early and uncivilized races. Among the
Egyptians the rising of the Nile served as a mark. The New
Zealanders were found to begin their year from the reappearance of
the Pleiades above the sea. One of the uses ascribed to birds, by the
Greeks, was to indicate the seasons by their migrations. Barrow
describes the aboriginal Hottentot as expressing dates by the
number of moons before or after the ripening of one of his chief
articles of food. He further states that the Kaffir chronology is kept
by the moon, and is registered by notches on sticks—the death of a
favourite chief, or the gaining of a victory, serving for a new era. By
which last fact, we are at once reminded that in early history, events
are commonly recorded as occurring in certain reigns, and in certain
years of certain reigns: a proceeding which made a king’s reign {46}
a rude measure of duration. And, as further illustrating the tendency
to divide time by natural phenomena and natural events, it may be
noticed that even by our own peasantry the definite divisions of
months and years are but little used; and that they habitually refer
to occurrences as “before sheep-shearing,” or “after harvest,” or
“about the time when the squire died.” It is manifest, therefore, that
the approximately equal periods perceived in Nature gave the first
units of measure for time; as did Nature’s approximately equal
lengths and weights give the first units of measure for space and
force.

It remains only to observe, that measures of value were similarly
derived. Barter, in one form or other, is found among all but the very
lowest human races. It is obviously based upon the notion of
equality of worth. And as it gradually merges into trade by the
introduction of some kind of currency, we find that the measures of
worth, constituting this currency, are organic bodies; in some cases
cowries, in others cocoa-nuts, in others cattle, in others pigs;
among the American Indians peltry or skins, and in Iceland dried
fish.
Notions of exact equality and of measure having been reached,
there arose definite ideas of magnitudes as being multiples one of
another; whence the practice of measurement by direct apposition
of a measure. The determination of linear extensions by this process
can scarcely be called science, though it is a step towards it; but the
determination of lengths of time by an analogous process may be
considered as one of the earliest samples of quantitative prevision.
For when it is first ascertained that the moon completes the cycle of
her changes in about thirty days—a fact known to most uncivilized
tribes that can count beyond the number of their fingers—it is
manifest that it becomes possible to say in what number of days any
specified phase of the moon will recur; and it is also manifest that
this prevision is effected by an apposition of two times, after the
same manner {47} that linear space is measured by the apposition of
two lines. For to express the moon’s period in days, is to say how
many of these units of measure are contained in the period to be
measured—is to ascertain the distance between two points in time
by means of a scale of days, just as we ascertain the distance
between two points in space by a scale of feet or inches; and in
each case the scale coincides with the thing measured—mentally in
the one, visibly in the other. So that in this simplest, and perhaps
earliest case of quantitative prevision, the phenomena are not only
thrust daily upon men’s notice, but Nature is, as it were, perpetually

repeating that process of measurement by observing which the
prevision is effected.
This fact, that in very early stages of social progress it is known
that the moon goes through her changes in nearly thirty days, and
that in rather more than twelve moons the seasons return—this fact
that chronological astronomy assumes a certain scientific character
even before geometry does; while it is partly due to the
circumstance that the astronomical divisions, day, month, and year,
are ready made for us, is partly due to the further circumstances
that agricultural and other operations were at first regulated
astronomically, and that from the supposed divine nature of the
heavenly bodies their motions determined the periodical religious
festivals. As instances of the one we have the observation of the
Egyptians, that the rising of the Nile corresponded with the heliacal
rising of Sirius; the directions given by Hesiod for reaping and
ploughing, according to the positions of the Pleiades; and his maxim
that “fifty days after the turning of the sun is a seasonable time for
beginning a voyage.” As instances of the other, we have the naming
of the days after the sun, moon, and planets; the early attempts
among Eastern nations to regulate the calendar so that the gods
might not be offended by the displacement of their sacrifices; and
the fixing of the great annual festival of the Peruvians by the
position of the sun. {48} In all which facts we see that, at first,
science was simply an appliance of religion and industry.
After the discoveries that a lunation occupies nearly thirty days,
and that some twelve lunations occupy a year—discoveries which we
may infer were the earliest, from the fact that existing uncivilized
races have made them—we come to the first known astronomical
records, which are those of eclipses. The Chaldeans were able to
predict these. “This they did, probably,” says Dr. Whewell in his
useful history, from which most of the materials we are about to use
will be drawn, “by means of their cycle of 223 months, or about
eighteen years; for, at the end of this time, the eclipses of the moon

begin to return, at the same intervals and in the same order as at
the beginning.” Now this method of calculating eclipses by means of
a recurring cycle,—the Saros as they called it—is a more complex
case of prevision by means of coincidence of measures. For by what
observations must the Chaldeans have discovered this cycle?
Obviously, as Delambre infers, by inspecting their registers; by
comparing the successive intervals; by finding that some of the
intervals were alike; by seeing that these equal intervals were
eighteen years apart; by discovering that all the intervals that were
eighteen years apart were equal; by ascertaining that the intervals
formed a series which repeated itself, so that if one of the cycles of
intervals were superposed on another the divisions would fit. And
this being once perceived, it became possible to use the cycle as a
scale of time by which to measure out future periods of recurrence.
Seeing thus that the process of so predicting eclipses, is in essence
the same as that of predicting the moon’s monthly changes by
observing the number of days after which they repeat—seeing that
the two differ only in the extent and irregularity of the intervals; it is
not difficult to understand how such an amount of knowledge should
so early have been reached. And we shall be the less surprised on
remembering that the only things involved in these {49} previsions
were time and number; and that the time was in a manner self-
numbered.
Still, the ability to predict events recurring only after so long a
period as eighteen years, implies a considerable advance in
civilization—a considerable development of general knowledge; and
we have now to inquire what progress in other sciences
accompanied, and was necessary to, these astronomical previsions.
In the first place, there must have been a tolerably efficient system
of calculation. Mere finger-counting, mere head-reckoning, even with
the aid of a decimal notation, could not have sufficed for numbering
the days in a year; much less the years, months, and days between
eclipses. Consequently there must have been a mode of registering

numbers; probably even a system of numerals. The earliest
numerical records, if we may judge by the practices of the less
civilized races now existing, were probably kept by notches cut on
sticks, or strokes marked on walls; much as public-house scores are
kept now. And there is reason to think that the first numerals used
were simply groups of straight strokes, as some of the still-extant
Roman ones are; leading us to suspect that these groups of strokes
were used to represent groups of fingers, as the groups of fingers
had been used to represent groups of objects—a supposition
harmonizing with the aboriginal practice of picture writing. Be this so
or not, however, it is manifest that before the Chaldeans discovered
their Saros, they must have had both a set of written symbols
serving for an extensive numeration, and a familiarity with the
simpler rules of arithmetic.
Not only must abstract mathematics have made some progress,
but concrete mathematics also. It is scarcely possible that the
buildings belonging to this era should have been laid out and erected
without any knowledge of geometry. At any rate, there must have
existed that elementary geometry which deals with direct {50}
measurement—with the apposition of lines; and it seems that only
after the discovery of those simple proceedings, by which right
angles are drawn, and relative positions fixed, could so regular an
architecture be executed. In the case of the other division of
concrete mathematics—mechanics, we have definite evidence of
progress. We know that the lever and the inclined plane were
employed during this period: implying that there was a qualitative
prevision of their effects, if not a quantitative one. But we know
more. We read of weights in the earliest records; and we find
weights in ruins of the highest antiquity. Weights imply scales, of
which we have also mention; and scales involve the primary theorem
of mechanics in its least complicated form—involve not a qualitative
but a quantitative prevision of mechanical effects. And here we may
notice how mechanics, in common with the other exact sciences,

took its rise from the simplest application of the idea of equality. For
the mechanical proposition which the scales involve, is, that if a
lever with equal arms, have equal weights suspended from them,
the weights will remain at equal altitudes. And we may further
notice how, in this first step of rational mechanics, we see illustrated
the truth awhile since named, that as magnitudes of linear extension
are the only ones of which the equality is exactly ascertainable, the
equalities of other magnitudes have at the outset to be determined
by means of them. For the equality of the weights which balance
each other in scales, depends on the equality of the arms: we can
know that the weights are equal only by proving that the arms are
equal. And when by this means we have obtained a system of
weights,—a set of equal units of force and definite multiples of
them, then does a science of mechanics become possible. Whence,
indeed, it follows, that rational mechanics could not possibly have
any other starting-point than the scales.
Let us further remember that during this same period {51} there
was some knowledge of chemistry. Sundry of the arts which we
know to have been carried on, were made possible only by a
generalized experience of the modes in which certain bodies affect
each other under special conditions. In metallurgy, which was
extensively practised, this is abundantly illustrated. And we even
have evidence that in some cases the knowledge possessed was, in
a sense, quantitative. For, as we find by analysis that the hard alloy
of which the Egyptians made their cutting tools, was composed of
copper and tin in fixed proportions, there must have been an
established prevision that such an alloy was to be obtained only by
mixing them in these proportions. It is true, this was but a simple
empirical generalization; but so was the generalization respecting
the recurrence of eclipses; so are the first gen
er
al
i
za
tions of every
science.
Respecting the simultaneous advance of the sciences during this
early epoch, it remains to point out that even the most complex of

them must have made some progress. For under what conditions
only were the foregoing developments possible? The conditions
furnished by an established and organized social system. A long
continued registry of eclipses; the building of palaces; the use of
scales; the practice of metallurgy—alike imply a settled and populous
nation. The existence of such a nation not only presupposes laws
and some administration of justice, which we know existed, but it
presupposes successful laws—laws conforming in some degree to
the conditions of social stability—laws enacted because it was found
that the actions forbidden by them were dangerous to the State. We
do not by any means say that all, or even the greater part, of the
laws were of this nature; but we do say, that the fundamental ones
were. It cannot be denied that the laws affecting life and property
were such. It cannot be denied that, however little these were
enforced between class and class, they were to a considerable
extent {52} enforced between members of the same class. It can
scarcely be questioned, that the administration of them between
members of the same class was seen by rulers to be necessary for
keeping society together. But supposition aside, it is clear that the
habitual recognition of these claims in their laws, implied some
prevision of social phenomena. That same idea of equality, which,
as we have seen, underlies other science, underlies also morals and
sociology. The conception of justice, which is the primary one in
morals; and the administration of justice, which is the vital condition
to social existence; are impossible without the recognition of a
certain likeness in men’s claims, in virtue of their common humanity.
Equity literally means equalness; and if it be admitted that there
were even the vaguest ideas of equity in these primitive eras, it must
be admitted that there was some appreciation of the equalness of
men’s liberties to pursue the objects of life—some appreciation,
therefore, of the essential principle of national equilibrium.
Thus in this initial stage of the positive sciences, before geometry
had yet done more than evolve a few empirical rules—before

mechanics had passed beyond its first theorem—before astronomy
had advanced from its merely chronological phase into the
geometrical; the most involved of the sciences had reached a certain
degree of development—a development without which no progress
in other sciences was possible.
Only noting as we pass, how, thus early, we may see that the
progress of exact science was not only towards an increasing
number of previsions, but towards previsions more accurately
quantitative—how, in astronomy, the recurring period of the moon’s
motions was by and by more correctly ascertained to be two
hundred and thirty-five lunations; how Callipus further corrected this
Metonic cycle, by leaving out a day at the end of every seventy-six
years; and how these successive advances implied a {53} longer
continued registry of observations, and the co-ordination of a
greater number of facts; let us go on to inquire how geometrical
astronomy took its rise. The first astronomical instrument was the
gnomon. This was not only early in use in the East, but it was found
among the Mexicans; the sole astronomical observations of the
Peruvians were made by it; and we read that 1100 B.C., the Chinese
observed that, at a certain place, the length of the sun’s shadow, at
the summer solstice, was to the height of the gnomon, as one and a
half to eight. Here again it is observable, both that the instrument is
found ready made, and that Nature is perpetually performing the
process of measurement. Any fixed, erect object—a column, a pole,
the angle of a building—serves for a gnomon; and it needs but to
notice the changing position of the shadow it daily throws, to make
the first step in geometrical astronomy. How small this first step was,
may be seen in the fact that the only things ascertained at the
outset were the periods of the summer and winter solstices, which
corresponded with the least and greatest lengths of the mid-day
shadow; and to fix which, it was needful merely to mark the point to
which each day’s shadow reached. And now let it not be overlooked
that in the observing at what time during the next year this extreme

limit of the shadow was again reached, and in the inference that the
sun had then arrived at the same turning point in his annual course,
we have one of the simplest instances of that combined use of equal
magnitudes and equal relations, by which all exact science, all
quantitative prevision, is reached. For the relation observed was
between the length of the gnomon’s shadow and the sun’s position
in the heavens; and the inference drawn was that when, next year,
the extremity of the shadow came to the same point, he occupied
the same place. That is, the ideas involved were, the equality of the
shadows, and the equality of the relations between {54} shadow and
sun in successive years. As in the case of the scales, the equality of
relations here recognized is of the simplest order. It is not as those
habitually dealt with in the higher kinds of scientific reasoning, which
answer to the general type—the relation between two and three
equals the relation between six and nine; but it follows the type—the
relation between two and three equals the relation between two and
three: it is a case of not simply equal relations, but coinciding
relations. And here, indeed, we may see beautifully illustrated how
the idea of equal relations takes its rise after the same manner that
that of equal magnitudes does. As already shown, the idea of equal
magnitudes arose from the observed coincidence of two lengths
placed together; and in this case we have not only two coincident
lengths of shadows, but two coincident relations between sun and
shadows.
From the use of the gnomon there naturally grew up the
conception of angular measurements; and with the advance of
geometrical conceptions came the hemisphere of Berosus, the
equinoctial armil, the solstitial armil, and the quadrant of Ptolemy—
all of them employing shadows as indices of the sun’s position, but
in combination with angular divisions. It is out of the question for us
here to trace these details of progress. It must suffice to remark that
in all of them we may see that notion of equality of relations of a
more complex kind, which is best illustrated in the astrolabe, an

instrument which consisted “of circular rims, moveable one within
the other, or about poles, and contained circles which were to be
brought into the position of the ecliptic, and of a plane passing
through the sun and the poles of the ecliptic”—an instrument,
therefore, which represented, as by a model, the relative positions of
certain imaginary lines and planes in the heavens; which was
adjusted by putting these representative lines and planes into
parallelism with the celestial ones; and which depended for its use
on the perception that the relations among these {55} representative
lines and planes were equal to the relations among those
represented. We might go on to point out how the conception of the
heavens as a revolving hollow sphere, the explanation of the moon’s
phases, and indeed all the successive steps taken, involved this
same mental process. But we must content ourselves with referring
to the theory of eccentrics and epicycles, as a further marked
illustration of it. As first suggested, and as proved by Hipparchus to
afford an explanation of the leading irregularities in the celestial
motions, this theory involved the perception that the progressions,
retrogressions, and variations of velocity seen in the heavenly
bodies, might be reconciled with their assumed uniform movements
in circles, by supposing that the earth was not in the centre of their
orbits; or by supposing that they revolved in circles whose centres
revolved round the earth; or by both. The discovery that this would
account for the appearances, was the discovery that in certain
geometrical diagrams the relations were such, that the uniform
motion of points along curves conditioned in specified ways, would,
when looked at from a particular position, present analogous
irregularities; and the calculations of Hipparchus involved the belief
that the relations subsisting among these geometrical curves were
equal to the relations subsisting among the celestial orbits.
Leaving here these details of astronomical progress, and the
philosophy of it, let us observe how the relatively concrete science of
geometrical astronomy, having been thus far helped forward by the

development of geometry in general, reacted upon geometry, caused
it also to advance, and was again assisted by it. Hipparchus, before
making his solar and lunar tables, had to discover rules for
calculating the relations between the sides and angles of triangles—
trigonometry, a subdivision of pure mathematics. Further, the
reduction of the doctrine of the sphere to a quantitative form
needed for astronomical purposes, required the formation of a
spherical trigonometry, which {56} was also achieved by Hipparchus.
Thus both plane and spherical trigonometry, which are parts of the
highly abstract and simple science of extension, remained
undeveloped until the less abstract and more complex science of the
celestial motions had need of them. The fact admitted by M. Comte,
that since Descartes the progress of the abstract division of
mathematics has been determined by that of the concrete division,
is paralleled by the still more significant fact that even thus early the
progress of mathematics was determined by that of astronomy. And
here, indeed, we see exemplified the truth, which the subsequent
history of science frequently illustrates, that before any more
abstract division makes a further advance, some more concrete
division suggests the necessity for that advance—presents the new
order of questions to be solved. Before astronomy put before
Hipparchus the problem of solar tables, there was nothing to raise
the question of the relations between lines and angles: the subject-
matter of trigonometry had not been conceived.
Just incidentally noticing the circumstance that the epoch we are
describing witnessed the evolution of algebra, a comparatively
abstract division of mathematics, by the union of its less abstract
divisions, geometry and arithmetic (a fact proved by the earliest
extant samples of algebra, which are half algebraic, half geometric)
we go on to observe that during the era in which mathematics and
astronomy were thus advancing, rational mechanics made its second
step; and something was done towards giving a quantitative form to
hydrostatics, optics, and acoustics. In each case we shall see how

the idea of equality underlies all quantitative prevision; and in what
simple forms this idea is first applied.
As already shown, the first theorem established in mechanics was,
that equal weights suspended from a lever with equal arms would
remain in equilibrium. Archimedes discovered that a lever with
unequal arms was in {57} equilibrium when one weight was to its
arm as the other arm to its weight; that is—when the numerical
relation between one weight and its arm was equal to the numerical
relation between the other arm and its weight.
The first advance made in hydrostatics, which we also owe to
Archimedes, was the discovery that fluids press equally in all
directions; and from this followed the solution of the problem of
floating bodies; namely, that they are in equilibrium when the
upward and downward pressures are equal.
In optics, again, the Greeks found that the angle of incidence is
equal to the angle of reflection; and their knowledge reached no
further than to such simple deductions from this as their geometry
sufficed for. In acoustics they ascertained the fact that three strings
of equal lengths would yield the octave, fifth and fourth, when
strained by weights having certain definite ratios; and they did not
progress much beyond this. In the one of which cases we see
geometry used in elucidation of the laws of light; and in the other,
geometry and arithmetic made to measure certain phenomena of
sound.
While sundry sciences had thus reached the first stages of
quantitative prevision, others were progressing in qualitative
prevision. It must suffice just to note that some small gen
er
al
i
za
tions
were made respecting evaporation, and heat, and electricity, and
magnetism, which, empirical as they were, did not in that respect
differ from the first gen
er
al
i
za
tions of every science; that the Greek
physicians had made advances in physiology and pathology, which,
considering the great imperfection of our present knowledge, are by
no means to be despised; that zoology had been so far systematized

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