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Chapter 07 - Input/Output Technology
Cengage Learning Testing, Powered by Cognero Page 1
1. For video display, a pixel displays no light or light of a specific color and intensity.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 231
2. Image quality improves as dots per inch increases.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 231
3. Image quality improves as pixel size increases.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 231
4. On paper, pixel size corresponds to the smallest drop of ink that can be placed accurately on the page.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 231
5. Decades ago, printers adopted 1/32 of an inch as a standard pixel size.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 231
6. For people and computers, a printed character must exactly match a specific pixel map to be recognizable.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 231
7. Point size refers to characters’ width.
a. True
b. False

ANSWER: False
POINTS: 1
REFERENCES: 231
8. A monochrome display can display black, white, and many shades of gray in between, so it requires 8 bits per pixel.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 233
9. An IDL can represent image components as embedded fonts, vectors, curves and shapes, and embedded bitmaps.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 235
10. IDLs are a simple form of compression.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 237
11. LCD displays have less contrast than other flat panel displays because color filters reduce the total amount of light
passing through the front of the panel.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 242
12. Phosphors emit colored light in liquid crystal displays.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 242
13. Because plasma displays actively generate colored light near the display surface, they’re brighter and have a wider
viewing angle than LCDs.
a. True
b. False
ANSWER: True
POINTS: 1

REFERENCES: 242
14. OLED displays combine many of the best features of LCD and plasma displays.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 243
15. Impact technology began with dot matrix printers.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 244
16. Color laser output uses four separate print generators.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 247
17. An advantage of optical over mechanical mice is a lack of moving parts that can be contaminated with dust and dirt.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 248
18. Bar-code readers are typically used to track large numbers of inventory items, as in grocery store inventory and
checkout, package tracking, warehouse inventory control, and zip code routing for postal mail.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 250
19. Modern bar codes encode data in three dimensions.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 251
20. Character and text recognition is most accurate when text is printed in a single font and style, with all text oriented in

the same direction on the page.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 252
21. Error rates of 1-2% are common using OCR software with mixed-font text and even higher with handwritten text.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 252
22. A digital still camera captures and stores one image at a time.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 252
23. Moving image quality improves as the number of frames per second (fps) decreases.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 253
24. Typically, digital cameras capture 14 to 20 fps.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 253
25. Most portable data capture devices combine a keyboard, mark or bar-code scanner, and wireless connection to a wired
base station, cash register, or computer system.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 253
26. For sound reproduction that sounds natural to people, frequencies between 20 Hz and 20 KHz must be sampled at
least 96,000 times per second.
a. True

b. False
ANSWER: False
POINTS: 1
REFERENCES: 253
27. Sound varies by frequency (pitch) and intensity (loudness).
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 253
28. Continuous speech is a series of nonstop interconnected phonemes.
a. True
b. False
ANSWER: False
POINTS: 1
REFERENCES: 254
29. Phonemes sound similar when voiced repetitively by the same person.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 254
30. A significant advantage of MIDI is its compact storage format.
a. True
b. False
ANSWER: True
POINTS: 1
REFERENCES: 257
31. Each cell in the matrix representing one part of a digital image is called a ____.
a. bubble
b. pixel
c. Dot
d. Block
ANSWER: b
POINTS: 1
REFERENCES: 231
32. The ____ of a display is the number of pixels displayed per linear measurement unit.
a. resolution
b. refinement

c. accuracy
d. pitch
ANSWER: a
POINTS: 1
REFERENCES: 231
33. In the United States, resolution is generally stated in ____.
a. lines per inch
b. pixels per line
c. dots per inch
d. dots per millimeter
ANSWER: c
POINTS: 1
REFERENCES: 231
34. Written Western languages are based on systems of symbols called ____.
a. fonts
b. characters
c. types
d. schemes
ANSWER: b
POINTS: 1
REFERENCES: 231
35. A collection of characters of similar style and appearance is called a ____.
a. type
b. scheme
c. pitch
d. font
ANSWER: d
POINTS: 1
REFERENCES: 231
36. The number of distinct colors or gray shades that can be displayed is sometimes called the ____.
a. resolution
b. palette
c. range
d. chromatic depth
ANSWER: d
POINTS: 1
REFERENCES: 233
37. A(n) ____ is simply a table of colors.
a. palette
b. spectrum

c. RGB system
d. color scheme
ANSWER: a
POINTS: 1
REFERENCES: 233
38. ____ is a process that generates color approximations by placing small dots of different colors in an interlocking
pattern.
a. Merging
b. Banding
c. Dithering
d. Retracing
ANSWER: c
POINTS: 1
REFERENCES: 234
39. In graphics, a ____ is a line segment with a specific angle and length in relation to a point of origin.
a. course
b. vector
c. path
d. route
ANSWER: b
POINTS: 1
REFERENCES: 236
40. Components of a video controller include display generator circuitry, software stored in ROM, a video processor, and
____.
a. RAM
b. font tables
c. display pixels
d. secondary storage
ANSWER: a
POINTS: 1
REFERENCES: 239
41. Video display panels are connected to a ____ that’s connected to a port on the system bus or a dedicated video bus.
a. video buffer
b. video manager
c. video station
d. video controller
ANSWER: d
POINTS: 1
REFERENCES: 239
42. The number of refresh cycles per second is normally stated in hertz and called the ____.

a. refresh rate
b. pixel depth
c. resolution
d. scan rate
ANSWER: a
POINTS: 1
REFERENCES: 240
43. A(n) ____ display includes one or more transistors for every display pixel.
a. cathode ray tube
b. active matrix
c. liquid crystal
d. Passive matrix
ANSWER: b
POINTS: 1
REFERENCES: 242
44. ____ technology etches display pixels and the transistors and traces that control/illuminate them onto a glass substrate.
a. CRT
b. Neon
c. backlight
d. TFT
ANSWER: d
POINTS: 1
REFERENCES: 242
45. A ____ contains a matrix of liquid crystals sandwiched between two polarizing filter panels that block all light except
light approaching from a specific angle.
a. Plasma
b. CRT
c. liquid crystal display
d. light emitting diode
ANSWER: c
POINTS: 1
REFERENCES: 241
46. ____ displays use excited gas and phosphors to generate colored light.
a. Plasma
b. liquid crystal
c. light emitting diode
d. thin film transmission
ANSWER: a
POINTS: 1
REFERENCES: 242

47. Modern ____ displays achieve high-quality color display with organic compounds.
a. LCD
b. TFT
c. CRT
d. LED
ANSWER: d
POINTS: 1
REFERENCES: 243
48. OLED displays combine features from both LED and plasma display, including: ____.
a. thin, bright, and high power
b. thin, bright, and low power
c. thin, backlit, and high power
d. thin, backlit, and low power
ANSWER: b
POINTS: 1
REFERENCES: 243
49. A(n) ____ printer moves a print head containing a matrix of pins over the paper.
a. laser
b. dot matrix
c. inkjet
d. dye sublimation
ANSWER: b
POINTS: 1
REFERENCES: 244
50. A modern large format printer is a _____ printer that can print on wider-than normal rolls of paper.
a. laser
b. dye sublimation
c. inkjet
d. impact
ANSWER: c
POINTS: 1
REFERENCES: 247
51. A(n) ____ operates with an electrical charge and the attraction of ink to this charge.
a. laser printer
b. inkjet printer
c. impact printer
d. thermal printer
ANSWER: a
POINTS: 1
REFERENCES: 246

52. A ____ is a printer that generates line drawings on wide sheets or rolls of paper.
a. sublimation
b. thermal
c. line printer
d. plotter
ANSWER: d
POINTS: 1
REFERENCES: 247
53. Pointing devices can be used to enter drawings into a computer system or control the position of a(n) ____ on a
display device.
a. pointer
b. arrow
c. cursor
d. marker
ANSWER: c
POINTS: 1
REFERENCES: 248
54. Touch position sensing in a touch screen is usually based on ____.
a. resistance
b. flux
c. capacitance
d. inductance
ANSWER: c
POINTS: 1
REFERENCES: 249
55. ____ sensors capture input from special-purpose symbols placed on paper or the flat surfaces of 3D objects.
a. Dot and image
b. Mark and image
c. Mark and pattern
d. Image capture
ANSWER: c
POINTS: 1
REFERENCES: 250
56. A(n) ____ detects specific patterns of bars or boxes.
a. bar-code scanner
b. image scanner
c. dimensional scanner
d. linear scanner
ANSWER: a
POINTS: 1
REFERENCES: 250

57. Bar-code readers use ____that sweep a narrow beam back and forth across the bar code.
a. scanning LEDs
b. high-intensity lamps
c. high resolution CCDs
d. scanning lasers
ANSWER: d
POINTS: 1
REFERENCES: 250
58. PDF417 bar codes can hold around ____ of data.
a. 1 KB
b. 1 MB
c. 1 GB
d. 1 TB
ANSWER: a
POINTS: 1
REFERENCES: 251
59. A(n) ____ generates bitmap representations of printed images.
a. bar-code scanner
b. image scanner
c. optical scanner
d. visual scanner
ANSWER: c
POINTS: 1
REFERENCES: 251
60. ____ devices combine optical-scanning technology with a special-purpose processor or software to interpret bitmap
content.
a. Optical image recognition
b. Optical character recognition
c. Optical character reproduction
d. Optical image resolution
ANSWER: b
POINTS: 1
REFERENCES: 251
61. The process of converting analog sound waves to digital representation is called ____.
a. reducing
b. interpreting
c. sampling
d. transforming
ANSWER: c
POINTS: 1

REFERENCES: 253
62. A(n) ____ accepts a continuous electrical signal representing sound (such as microphone input), samples it at regular
intervals, and outputs a stream of bits representing the samples.
a. analog-to-digital converter
b. analog-to-digital inverter
c. analog-to-digital diverter
d. analog-to-digital parser
ANSWER: a
POINTS: 1
REFERENCES: 254
63. A(n) ____ accepts a stream of bits representing sound samples and generating a continuous electrical signal that can
be amplified and routed to a speaker.
a. digital-to-analog processor
b. digital-to-analog parser
c. digital-to-analog compiler
d. digital-to-analog converter
ANSWER: d
POINTS: 1
REFERENCES: 254
64. ____ output is only able to generate one frequency (note) at a time.
a. Stereophonic
b. Monophonic
c. Polyphonic
d. Monosyllabic
ANSWER: b
POINTS: 1
REFERENCES: 254
65. ____ is the process of recognizing and responding to the meaning embedded in spoken words, phrases, or sentences.
a. Text recognition
b. Pattern recognition
c. Speech recognition
d. Natural recognition
ANSWER: c
POINTS: 1
REFERENCES: 254
66. Human speech consists of a series of sounds called ____, roughly corresponding to the sounds of each letter of the
alphabet.
a. phonemes
b. homonyms
c. cheremes

d. visemes
ANSWER: a
POINTS: 1
REFERENCES: 254
67. Most current speech-recognition systems are ____, which means they must be “trained” to recognize the sounds of
human speakers.
a. speaker independent
b. speaker dependent
c. speaker neutral
d. speaker attuned
ANSWER: b
POINTS: 1
REFERENCES: 255
68. A(n) ____ is a microprocessor specialized for processing continuous streams of audio or graphical data.
a. analog signal processor
b. virtual signal processor
c. electronic signal processor
d. digital signal processor
ANSWER: d
POINTS: 1
REFERENCES: 256
69. ____ is a standard for storing and transporting control information between computers and electronic musical
instruments.
a. Musical Instrument Digital Interface
b. Musical Instrument Digital Interface
c. Musical Interface Digital Interconnection
d. Musical Interconnection Digital Interface
ANSWER: a
POINTS: 1
REFERENCES: 257
70. Up to ____ channels of MIDI data can be sent over the same serial transmission line
a. 4
b. 8
c. 16
d. 32
ANSWER: c
POINTS: 1
REFERENCES: 258
71. As ____________________ size increases image quality improves.
ANSWER: pixel

POINTS: 1
REFERENCES: 231
72. To an observer, the quality of a printed or displayed image increases as ____________________ size increases.
ANSWER: pixel
POINTS: 1
REFERENCES: 231
73. Font size is measured in units called ____________________.
ANSWER: points
POINTS: 1
REFERENCES: 231
74. The ____________________ colors are cyan, magenta, and yellow.
ANSWER: subtractive
POINTS: 1
REFERENCES: 233
75. A stored set of numbers describing the content of all pixels in an image is called a(n) ____________________.
ANSWER: bitmap
POINTS: 1
REFERENCES: 233
76. ____________________ dithering is usually called half-toning.
ANSWER: Grayscale
POINTS: 1
REFERENCES: 234
77. Postscript is a(n) ____________________ designed mainly for printed documents, although it can also be used to
generate video display outputs.
ANSWER: image description language (IDL)
POINTS: 1
REFERENCES: 237
78. Each transfer of a full screen of data from the display generator to the monitor is called a(n) ____________________.
ANSWER: refresh cycle
POINTS: 1
REFERENCES: 240
79. Direct3D and ____________________ are widely-used video controller IDLs.
ANSWER: OpenGL
POINTS: 1
REFERENCES: 240
80. A(n) ____________________ matrix display uses one or more transistors for every pixel.
ANSWER: active

POINTS: 1
REFERENCES: 242
81. A(n) ____________________ matrix display shares transistors among rows and columns of pixels.
ANSWER: passive
POINTS: 1
REFERENCES: 242
82. A(n) plasma display pixel excites gas into a(n) ____________________ plasma state to generate UV light.
ANSWER: plasma
POINTS: 1
REFERENCES: 242
83. Of all flat panel displays, ____________________ have the shortest operational lifetimes.
ANSWER: plasma displays
POINTS: 1
REFERENCES: 243
84. When keys are pressed, a keyboard controller generates output called a(n) ____________________.
ANSWER: scan code
POINTS: 1
REFERENCES: 248
85. A(n) mouse that can detect motion with ____________________ dimensions uses an embedded gyroscope.
ANSWER: three
POINTS: 1
REFERENCES: 249
86. A(n) ____________________ is an LCD or LED display with additional TFT layers that detect the position of
electrical field changes based on capacitance.
ANSWER: touchscreen
POINTS: 1
REFERENCES: 249
87. Digitizing tablets and tablet PCs are examples of ____________________, a general class of input devices.
ANSWER: input pads
POINTS: 1
REFERENCES: 260
88. ____________________ touchscreen input interprets a sequence of touch information as a single command
ANSWER: Gesture-based
POINTS: 1
REFERENCES: 249
89. A(n) ____________________ scans for light or dark marks at specific locations on a page.
ANSWER: mark sensor

POINTS: 1
REFERENCES: 250
90. Digital still cameras, video cameras, and Webcams use an array of ____________________ placed behind lenses to
capture reflected and focused ambient light.
ANSWER: photosensors
POINTS: 1
REFERENCES: 251
91. Human-assisted ____________________ procedures are required in many applications to deal with error rates of
OCR technology.
ANSWER: quality control
POINTS: 1
REFERENCES: 252
92. ____________________ is a series of phonemes interspersed with periods of silence.
ANSWER: Continuous speech
POINTS: 1
REFERENCES: 254
93. A device that generates spoken messages based on text input is called a(n) ____________________.
ANSWER: audio response unit
POINTS: 1
REFERENCES: 256
94. The term ____________________ describes hardware that can generate multiple sound frequencies at the same time.
ANSWER: polyphonic
POINTS: 1
REFERENCES: 254
95. Describe the relationship between image quality, pixel size, and resolution?
ANSWER: Resolution is the number of pixels per linear inch in a printed or displayed image. As resolution
increases, the amount of fine detail in the image also increases. People perceive images with greater
amounts of fine detail as being of higher quality.
POINTS: 1
REFERENCES: 231
96. Explain the term point size as it relates to fonts.
ANSWER: A point is 1/72 of an inch. A font’s point size is the number of points between the top of the highest
character and the bottom of the lowest character in the font.
POINTS: 1
REFERENCES: 231
97. Of liquid crystal, plasma, LED, and OLED displays, which is best and why?
ANSWER: If you ignore cost and operational lifetime then LOED displays have the better performance
characteristics than the other types. They’re the thinnest, are at least tied for the brightest, are at least tied
for the lowest power consumption, and have the best color accuracy. Adding cost and operational

lifetime into the comparison complicates it because OLED has shorter operational lifetimes than liquid
crystal and LED displays and because OLED displays are the most expensive of all types. Those
statements are accurate as of 2015 but they may not be a few years later because OLED technology is the
newest and the most rapidly improving.
POINTS: 1
REFERENCES: 241
243
98. Why do modern systems typically use 2D bar codes instead of more traditional bar codes composed of lines of varying
thickness.
ANSWER: 2D bar codes can store more data in the same amount of 2D space.
POINTS: 1
REFERENCES: 251
99. Describe the components of a sound card.
ANSWER: General-purpose audio hardware can be integrated on a PC motherboard or packaged as an expansion
card that connects to the system bus (commonly called a sound card). At minimum, sound cards include
an ADC, a DAC, a low-power amplifier, and connectors (jacks) for a microphone and a speaker or
headphones. More elaborate cards might include the following:
• Multichannel surround sound, such as Dolby 5.1
• A general-purpose Musical Instrument Digital Interface (MIDI) synthesizer
• MIDI input and output jacks
• A more powerful amplifier to accommodate larger speakers and generate more volume
POINTS: 1
REFERENCES: 256

Exploring the Variety of Random
Documents with Different Content

As regards the angle of retardation, β, in the formula
r
′ = λe
θ cot α
, or r
′ = e
(θ − β)cot α
,
and in the case
r
′ = e
(2π − β)cot α
, or −log λ = (2π − β)cot α,
{546}
it is evident that when β = 2π, that will mean that λ = 1. In other
words, the outer and inner borders of the tube are identical, and the
tube is constituted by one continuous line.
When λ is a very small fraction, that is to say when the rates of
growth of the two borders of the tube are very diverse, then β will
tend towards infinity—tend that is to say towards a condition in
which the inner border of the tube never grows at all. This condition
is not infrequently approached in nature. The nearly parallel-sided
cone of Dentalium, or the widely separated whorls of Lituites, are
evidently cases where λ nearly approaches unity in the one case,
and is still large in the other, β being correspondingly small; while we
can easily find cases where β is very large, and λ is a small fraction,
for instance in Haliotis, or in Gryphaea.
For the purposes of the morphologist, then, the main result of
this last general in
ves
ti
ga
tion is to shew that all the various types of
“open” and “closed” spirals, all the various degrees of separation or
overlap of the successive whorls, are simply the outward expression
of a varying ratio in the rate of growth of the outer as compared
with the inner border of the tubular shell.
The foregoing problem of contact, or intersection, of the
successive whorls, is a very simple one in the case of the discoid
shell but a more complex one in the turbinate. For in the discoid
shell contact will evidently take place when the retardation of the
inner as compared with the outer whorl is just 360°, and the shape
of the whorls need not be considered.
As the angle of retardation diminishes from 360°, the whorls will
stand further and further apart in an open coil; as it increases
beyond 360°, they will more and more overlap; and when the angle
of retardation is infinite, that is to say when the true inner edge of

the whorl does not grow at all, then the shell is said to be
completely involute. Of this latter condition we have a striking
example in Argonauta, and one a little more obscure in Nautilus
pompilius.
In the turbinate shell, the problem of contact is twofold, for we
have to deal with the possibilities of contact on the same side of the
axis (which is what we have dealt with in the discoid) and {547} also
with the new possibility of contact or intersection on the opposite
side; it is this latter case which will determine the presence or
absence of an umbilicus, and whether, if present, it will be an open
conical space or a twisted cone. It is further obvious that, in the
case of the turbinate, the question of contact or no contact will
depend on the shape of the generating curve; and if we take the
simple case where this generating curve may be considered as an
ellipse, then contact will be found to depend on the angle which the
major axis of this ellipse makes with the axis of the shell. The
question becomes a complicated one, and the student will find it
treated in Blake’s paper already referred to.
When one whorl overlaps another, so that the generating curve
cuts its predecessor (at a distance of 2π) on the same radius vector,
the locus of intersection will follow a spiral line upon the shell, which
is called the “suture” by conchologists. It is evidently one of that
ensemble of spiral lines in space of which, as we have seen, the
whole shell may be conceived to be constituted; and we might call it
a “contact-spiral,” or “spiral of intersection.” In discoid shells, such as
an Ammonite or a Planorbis, or in Nautilus umbilicatus, there are
obviously two such contact-spirals, one on each side of the shell,
that is to say one on each side of a plane perpendicular to the axis.
In turbinate shells such a condition is also possible, but is somewhat
rare. We have it for instance, in Solarium perspectivum, where the
one contact-spiral is visible on the exterior of the cone, and the
other lies internally, winding round the open cone of the
umbilicus521; but this second contact-spiral is usually imaginary, or
concealed within the whorls of the turbinated shell. Again, in
Haliotis, one of the contact-spirals is non-existent, because of the

extreme obliquity of the plane of the generating curve. In Scalaria
pretiosa and in Spirula there is no contact-spiral, because the
growth of the generating curve has been too slow, in comparison
with the vector rotation of its plane. In Argonauta and in Cypraea,
there is no contact-spiral, because the growth of the generating
curve has been too quick. Nor, of course, is there any contact-spiral
in Patella or in Dentalium, because the angle α is too small ever to
give us a complete revolution of the spire. {548}
The various forms of straight or spiral shells among the
Cephalopods, which we have seen to be capable of complete
definition by the help of elementary mathematics, have received a
very complicated descriptive nomenclature from the
palaeontologists. For instance, the straight cones are spoken of as
orthoceracones or bactriticones, the loosely coiled forms as
gyroceracones or mimoceracones, the more closely coiled shells, in
which one whorl overlaps the other, as nautilicones or
ammoniticones, and so forth. In such a succession of forms the
biologist sees undoubted and unquestioned evidence of ancestral
descent. For instance we read in Zittel’s Palaeontology522: “The
bactriticone obviously represents the primitive or primary radical of
the Ammonoidea, and the mimoceracone the next or secondary
radical of this order”; while precisely the opposite conclusion was
drawn by Owen, who supposed that the straight chambered shells of
such fossil cephalopods as Orthoceras had been produced by the
gradual unwinding of a coiled nautiloid shell523. To such
phylogenetic hypotheses the math
e
mat
i
cal or dynamical study of the
forms of shells lends no valid support. If we have two shells in which
the constant angle of the spire be respectively 80° and 60°, that fact
in itself does not at all justify an assertion that the one is more
primitive, more ancient, or more “ancestral” than the other. Nor, if
we find a third in which the angle happens to be 70°, does that fact
entitle us to say that this shell is intermediate between the other
two, in time, or in blood relationship, or in any other sense
whatsoever save only the strictly formal and math
e
mat
i
cal one. For
it is evident that, though these particular arithmetical constants

manifest themselves in visible and recognisable differences of form,
yet they are not necessarily more deep-seated or significant than are
those which manifest themselves only in difference of magnitude;
and the student of phylogeny scarcely ventures to draw conclusions
as to the relative antiquity of two allied organisms on the ground
that one happens to be bigger or less, or longer or shorter, than the
other. {549}
At the same time, while it is obviously unsafe to rest conclusions
upon such features as these, unless they be strongly supported and
corroborated in other ways,—for the simple reason that there is
unlimited room for coincidence, or separate and independent
attainment of this or that magnitude or numerical ratio,—yet on the
other hand it is certain that, in particular cases, the evolution of a
race has actually involved gradual increase or decrease in some one
or more numerical factors, magnitude itself included,—that is to say
increase or decrease in some one or more of the actual and relative
velocities of growth. When we do meet with a clear and
unmistakable series of such progressive magnitudes or ratios,
manifesting themselves in a progressive series of “allied” forms, then
we have the phenomenon of “orthogenesis.” For orthogenesis is
simply that phenomenon of continuous lines or series of form (and
also of functional or physiological capacity), which was the
foundation of the Theory of Evolution, alike to Lamarck and to
Darwin and Wallace; and which we see to exist whatever be our
ideas of the “origin of species,” or of the nature and origin of
“functional adaptations.” And to my mind, the math
e
mat
i
cal (as
distinguished from the purely physical) study of morphology bids fair
to help us to recognise this phenomenon of orthogenesis in many
cases where it is not at once patent to the eye; and also, on the
other hand, to warn us, in many other cases, that even strong and
apparently complex resemblances in form may be capable of arising
independently, and may sometimes signify no more than the equally
accidental numerical coincidences which are manifested in identity of
length or weight, or any other simple magnitudes.

I have already referred to the fact that, while in general a very
great and remarkable regularity of form is char
ac
ter
is
tic of the
molluscan shell, that complete regularity is apt to be departed from.
We have clear cases of such a departure in Pupa, Clausilia, and
various Bulimi, where the enveloping cone of the spire is not a right
cone but a cone whose sides are curved. It is plain that this
condition may arise in two ways: either by a gradual change in the
ratio of growth of the whorls, that is to say in the logarithmic spiral
itself, or by a change in the velocity of {550} translation along the axis,
that is to say in the helicoid which, in all turbinate shells, is
superposed upon the spiral. Very careful measurements will be
necessary to determine to which of these factors, or in what
proportions to each, the phenomenon is due. But in many
Ammonitoidea where the helicoid factor does not enter into the
case, we have a clear illustration of gradual and marked changes in
the spiral angle itself, that is to say of the ratio of growth cor
re
spon
‐
ding to increase of vectorial angle. We have seen from some of
Naumann’s and Grabau’s measurements that such a tendency to
vary, such an acceleration or retardation, may be detected even in
Ammonites which present nothing abnormal to the eye. But let us
suppose that the spiral angle increases somewhat rapidly; we shall
then get a spiral with gradually narrowing whorls, and this condition
is char
ac
ter
is
tic
Fig. 281. An ammonitoid shell (Macroscaphites) to shew
change of curvature.

of Oekotraustes, a subgenus of Ammonites. If on the other hand,
the angle α gradually diminishes, and even falls away to zero, we
shall have the spiral curve opening out, as it does in Scaphites,
Ancyloceras and Lituites, until the spiral coil is replaced by a spiral
curve so gentle as to seem all but straight. Lastly, there are a few
cases, such as Bellerophon expansus and some Goniatites, where
the outer spiral does not perceptibly change, but the whorls become
more “embracing” or the whole shell more involute. Here it is the
angle of retardation, the ratio of growth between the outer and
inner parts of the whorl, which undergoes a gradual change.
In order to understand the relation of a close-coiled shell to one
of its straighter congeners, to compare (for example) an {551}
Ammonite with an Orthoceras, it is necessary to estimate the length
of the right cone which has, so to speak, been coiled up into the
spiral shell. Our problem then is, To find the length of a plane
logarithmic spiral, in terms of the radius and the constant angle α. In
the annexed diagram, if OP be a radius vector, OQ a line of
reference perpendicular to OP, and PQ a tangent to the curve, PQ,
or sec α, is equal in length to the spiral arc OP. And this is practically
obvious: for PP
′ ⁄ PR
′ = ds ⁄ dr = sec α, and therefore sec α = s ⁄ r, or
the ratio of arc to radius vector.
Fig. 282.

Accordingly, the ratio of l, the total length, to r, the radius vector
up to which the total length is to be measured, is expressed by a
simple table of secants; as follows:

α l ⁄ r
5° 1·004
10 1·015
20 1·064
30 1·165
40 1·305
50 1·56
60 2·0
70 2·9
75 3·9
80 5·8
85 11·5
86 14·3
87 19·1
88 28·7
89 57·3
89° 10
′ 68·8
20 85·9
30 114·6
40 171·9
50 343·8
55 687·5
59 3437·7
90 Infinite
Putting the same table inversely, so as to shew the total {552}
length in whole numbers, in terms of the radius, we have as follows:

Total length
(in terms of
the radius)
Constant angle
2 60°
3 70 31
′
4 75 32
5 78 28
10 84 16
20 87 8
30 88 6
40 88 34
50 88 51
100 89 26
1000 89 56
′ 36
″
10,000 89 59 30
Accordingly, we see that (1), when the constant angle of the
spiral is small, the spiral itself is scarcely distinguishable from a
straight line, and its length is but very little greater than that of its
own radius vector. This remains pretty much the case for a
considerable increase of angle, say from 0° to 20° or more; (2) for a
very considerably greater increase of the constant angle, say to 50°
or more, the shell would only have the appearance of a gentle
curve; (3) the char
ac
ter
is
tic close coils of the Nautilus or Ammonite
would be typically represented only when the constant angle lies
within a few degrees on either side of about 80°. The coiled up
spiral of a Nautilus, with a constant angle of about 80°, is about six
times the length of its radius vector, or rather more than three times
its own diameter; while that of an Ammonite, with a constant angle
of, say, from 85° to 88°, is from about six to fifteen times as long as
its own diameter. And (4) as we approach an angle of 90° (at which

point the spiral vanishes in a circle), the length of the coil increases
with enormous rapidity. Our spiral would soon assume the
appearance of the close coils of a Nummulite, and the successive
increments of breadth in the successive whorls would become
inappreciable to the eye. The logarithmic spiral of high constant
angle would, as we have already seen, tend to become in
dis
tin
guish
‐
able, without the most careful measurement, from an Archimedean
spiral. And it is obvious, moreover, that our ordinary methods of {553}
determining the constant angle of the spiral would not in these cases
be accurate enough to enable us to measure the length of the coil:
we should have to devise a new method, based on the measurement
of radii or diameters over a large number of whorls.
The geometrical form of the shell involves many other beautiful
properties, of great interest to the mathematician, but which it is not
possible to reduce to such simple expressions as we have been
content to use. For instance, we may obtain an equation which shall
express completely the surface of any shell, in terms of polar or of
rectangular coordinates (as has been done by Moseley and by
Blake), or in Hamiltonian vector notation. It is likewise possible
(though of little interest to the naturalist) to determine the area of a
conchoidal surface, or the volume of a conchoidal solid, and to find
the centre of gravity of either surface or solid524. And Blake has
further shewn, with considerable elaboration, how we may deal with
the symmetrical distortion, due to pressure, which fossil shells are
often found to have undergone, and how we may reconstitute by
calculation their original undistorted form,—a problem which, were
the available methods only a little easier, would be very helpful to
the palaeontologist; for, as Blake himself has shewn, it is easy to
mistake a symmetrically distorted specimen of (for instance) an
Ammonite, for a new and distinct species of the same genus. But it
is evident that to deal fully with the math
e
mat
i
cal problems
contained in, or suggested by, the spiral shell, would require a whole
treatise, rather than a single chapter of this elementary book. Let us
then, leaving mathematics aside, attempt to summarise, and

perhaps to extend, what has been said about the general
possibilities of form in this class of organisms.

The Univalve Shell: a summary.
The surface of any shell, whether discoid or turbinate, may be
imagined to be generated by the revolution about a fixed axis of a
closed curve, which, remaining always geometrically similar to itself,
increases continually its dimensions: and, since the rate of growth of
the generating curve and its velocity of rotation follow the same law,
the curve traced in space by cor
re
spon
ding points {554} in the
generating curve is, in all cases, a logarithmic spiral. In discoid shells,
the generating figure revolves in a plane perpendicular to the axis, as
in Nautilus, the Argonaut and the Ammonite. In turbinate shells, it
slides continually along the axis of revolution, and the curve in space
generated by any given point partakes, therefore, of the character of a
helix, as well as of a logarithmic spiral; it may be strictly entitled a
helico-spiral. Such turbinate or helico-spiral shells include the snail, the
periwinkle and all the common typical Gastropods.
The generating figure, as represented by the mouth of the shell, is
sometimes a plane curve, of simple form; in other and more numerous
cases, it becomes more complicated in form and its boundaries do not
lie in one plane: but in such cases as these we
may replace it by its “trace,” on a plane at some definite angle to the
direction of growth, for instance by its form as it appears in a section
through the axis of the heli
coid shell. The gen
er
ating curve is of very
various shapes. It is circular in Scalaria or Cyclostoma, and in Spirula;
it may be con
si
dered as a seg
ment of a circle in Natica or in Plan
orbis.
It is ap
prox
i
mate
ly tri
an
gular in Conus, and rhom
boidal in Solarium or
Potam
ides. It is very com
monly more or less elliptical: the long axis of
the el
lipse being parallel to the axis of the shell in Oliva and Cypraea;
all but per
pen
di
cu
lar to it in many Trochi; and oblique to it in many
well-marked cases, such as Sto
ma
tella, La
mel
laria, Si
gar
etus hal
io
‐
toides (Fig. 284) and Haliotis. In Nautilus pom
pi
lius it is ap
prox
i
mate
ly
a semi-ellipse, and in N. um
bil
i
catus rather more than a semi-ellipse,
the long axis lying in both cases per
pen
di
cu
lar to the axis of the
shell525. Its {555} form is seldom open to easy math
e
mat
i
cal ex
pres
sion,
save when it is an actual circle or ellipse; but an exception to this rule
may be found in certain Am
mo
nites, for
ming the group “Cordati,”

Fig. 283. Section of a spiral, or
turbinate, univalve, Triton corrugatus,
Lam. (From Woodward.)
where (as Blake points out) the
curve is very nearly rep
re
sent
ed by
a cardioid, whose equation is r
= a(1 + cos θ).
The generating curve may grow slowly or quickly; its growth-factor
is very slow in Dentalium or Turritella, very rapid in Nerita, or Pileopsis,
or Haliotis or the Limpet. It may contain the axis in its plane, as in
Nautilus; it may be parallel to the axis, as in the majority of
Gastropods; or it may be inclined to the axis, as it is in a very marked
degree in Haliotis. In fact, in Haliotis the generating curve is so oblique
to the axis of the shell that the latter appears to grow by additions to
one margin only (cf. Fig. 258), as in the case of the opercula of Turbo

and Nerita referred to on p. 522; and this is what Moseley supposed it
to do.
Fig. 284. A, Lamellaria perspicua; B, Sigaretus haliotoides.
(After Woodward.)
The general appearance of the entire shell is determined (apart
from the form of its generating curve) by the magnitude of three
angles; and these in turn are determined, as has been sufficiently
explained, by the ratios of certain velocities of growth. These angles
are (1) the constant angle of the logarithmic spiral (α); (2) in turbinate
shells, the enveloping angle of the cone, or (taking half that angle) the
angle (θ) which a tangent to the whorls makes with the axis of the
shell; and (3) an angle called the “angle of retardation” (β), which
expresses the retardation in growth of {556} the inner as compared with
the outer part of each whorl, and therefore measures the extent to
which one whorl overlaps, or the extent to which it is separated from,
another.
The spiral angle (α) is very small in a limpet, where it is usually
taken as = 0°; but it is evidently of a significant amount, though
obscured by the shortness of the tubular shell. In Dentalium it is still
small, but sufficient to give the appearance of a regular curve; it
amounts here probably to about 30° to 40°. In Haliotis it is from about
70° to 75°; in Nautilus about 80°; and it lies between 80° and 85°, or
even more, in the majority of Gastropods.
The case of Fissurella is curious. Here we have, apparently, a
conical shell with no trace of spiral curvature, or (in other words) with
a spiral angle which approximates to 0°; but in the minute embryonic
shell (as in that of the limpet) a spiral convolution is distinctly to be
seen. It would seem, then, that what we have to do with here is an

unusually large growth-factor in the generating curve, which causes
the shell to dilate into a cone of very wide angle, the apical portion of
which has become lost or absorbed, and the remaining part of which is
too short to show clearly its intrinsic curvature. In the closely allied
Emarginula, there is likewise a well-marked spiral in the embryo, which
however is still manifested in the curvature of the adult, nearly conical,
shell. In both cases we have to do with a very wide-angled cone, and
with a high retardation-factor for its inner, or posterior, border. The
series is continued, from the apparently simple cone to the complete
spiral, through such forms as Calyptraea.
The angle α, as we have seen, is not always, nor rigorously, a
constant angle. In some Ammonites it may increase with age, the
whorls becoming closer and closer; in others it may decrease rapidly,
and even fall to zero, the coiled shell then straightening out, as in
Lituites and similar forms. It diminishes somewhat, also, in many
Orthocerata, which are slightly curved in youth, but straight in age. It
tends to increase notably in some common land-shells, the Pupae and
Bulimi; and it decreases in Succinea.
Directly related to the angle α is the ratio which subsists between
the breadths of successive whorls. The following table gives a few
illustrations of this ratio in particular cases, in addition to those which
we have already studied. {557}

Ratio of breadth of consecutive whorls.
Pointed Turbinates Obtuse Turbinates and Discoids
Telescopium fuscum 1·14 Conus virgo 1·25
Acus subulatus 1·16 Conus litteratus 1·40
*Turritella terebellata 1·18 Conus betulina 1·43
*Turritella imbricata 1·20 *Helix nemoralis 1·50
Cerithium palustre 1·22 *Solarium perspectivum 1·50
Turritella duplicata 1·23 Solarium trochleare 1·62
Melanopsis terebralis 1·23 Solarium magnificum 1·75
Cerithium nodulosum 1·24 *Natica aperta 2·00
*Turritella carinata 1·25 Euomphalus pentangulatus 2·00
Acus crenulatus 1·25 Planorbis corneas 2·00
Terebra maculata (Fig. 285) 1·25 Solaropsis pellis-serpentis 2·00
*Cerithium lignitarum 1·26 Dolium zonatum 2·10
Acus dimidiatus 1·28 *Natica glaucina 3·00
Cerithium sulcatum 1·32 Nautilus pompilius 3·00
Fusus longissimus 1·34 Haliotis excavatus 4·20
*Pleurotomaria conoidea 1·34 Haliotis parvus 6·00
Trochus niloticus (Fig. 286) 1·41 Delphinula atrata 6·00
Mitra episcopalis 1·43 Haliotis rugoso-plicata 9·30
Fusus antiquus 1·50 Haliotis viridis 10·00
Scalaria pretiosa 1·56
Fusus colosseus 1·71
Phasianella bulloides 1·80
Helicostyla polychroa 2·00
Those marked * from Naumann; the rest from Macalister
526.
In the case of turbinate shells, we require to take into account the
angle θ, in order to determine the spiral angle α from the ratio of the
breadths of consecutive whorls; for the short table given on p. 534 is
only applicable to discoid shells, in which the angle θ is an angle of
90°. Our formula, as mentioned on p. 518 now becomes
R = ε
2π sin θ cot α
.
For this formula I have worked out the following table. {558}

Table shewing values of the spiral angle α cor
re
spon
ding to certain ratios of breadth of
successive whorls of the shell, for various values of the apical semi-angle θ.
Ratio
R ⁄ 1
θ=5° 10° 15° 20° 30° 40° 50° 60° 70° 80° 90°
1·1 80° 8
′ 85° 0
′ 86° 44
′ 87° 28
′ 88° 16
′ 88° 39
′ 88° 52
′ 89° 0
′ 89° 4
′ 89° 7
′ 89° 8
′
1·25 67 51 78 27 82 11 84 5 85 56 86 50 87 21 87 39 87 50 87 56 87 58
1·5 53 30 69 37 76 0 79 21 82 39 84 16 85 13 85 44 86 4 86 15 86 18
2·0 38 20 57 35 66 55 73 11 77 34 80 16 81 52 82 45 83 18 83 37 83 42
2·5 30 53 50 0 60 35 67 0 73 45 77 13 79 19 80 26 81 11 81 35 81 42
3·0 26 32 44 50 56 0 63 0 70 45 74 45 77 17 78 35 79 28 79 56 80 5
3·5 23 37 41 5 52 25 59 50 68 15 72 45 75 35 77 2 78 1 78 33 78 43
4·0 21 35 38 10 49 35 57 15 66 10 71 3 74 9 75 42 76 47 77 22 77 34
4·5 20 0 36 0 47 15 55 5 64 25 69 35 72 54 74 33 75 43 76 20 76 35
5·0 18 45 34 10 45 20 53 15 62 55 68 15 71 48 73 31 74 45 75 25 75 38
10·0 13 25 25 20 35 15 43 5 53 45 60 20 64 57 67 4 68 42 69 35 69 53
20·0 10 25 20 0 28 30 35 45 46 25 53 25 58 52 61 10 63 6 64 10 64 31
50·0 8 0 15 35 22 35 28 50 38 45 45 55 52 1 54 18 56 28 57 42 58 6
100·0 6 50 13 20 19 30 25 5 34 20 41 15 47 35 49 45 52 3 53 20 53 46
{559}
From this table, by interpolation, we may easily fill in the ap
prox
i
‐
mate values of α, as soon as we have determined the apical angle θ
and measured the ratio R; as follows:

R θ α
Turritella sp. 1·12 7° 81°
Cerithium nodulosum 1·24 15 82
Conus virgo 1·25 70 88
Mitra episcopalis 1·43 16 78
Scalaria pretiosa 1·56 26 81
Phasianella bulloides 1·80 26 80
Solarium perspectivum 1·50 53 85
Natica aperta 2·00 70 83
Planorbis corneus 2·00 90 84
Euomphalus pentangulatus 2·00 90 84
We see from this that shells so different in appearance as
Cerithium, Solarium, Natica and Planorbis differ very little indeed in the
magnitude of the spiral angle α, that is to say in the relative velocities
of radial and tangential growth. It is upon the angle θ
that the difference in their form mainly depends: that is to say the
amount of longitudinal shearing, or displacement parallel to the axis of
the shell.
The enveloping angle, or rather semi-angle (θ), of the cone may be
taken as 90° in the discoid shells, such as Nautilus and Planorbis. It is
still a large angle, of 70° or 75°, in Conus or in Cymba, somewhat less
in Cassis, Harpa, Dolium or Natica; it is about 50° to 55° in the various
species of Solarium, about 35° in the typical Trochi, such as T.
niloticus or T. zizyphinus, and about 25° or 26° in Scalaria pretiosa
and Phasianella bulloides; it becomes a very acute angle, of 15°, 10°,
or even less, in Eulima, Turritella or Cerithium. The costly Conus gloria-
maris, one of the {560} great treasures of the conchologist, differs from
its congeners in no important particular save in the somewhat
“produced” spire, that is to say in the comparatively low value of the
angle θ.

Fig. 285. Terebra maculata, L.
A variation with advancing age of θ is common, but (as Blake
points out) it is often not to be distinguished or disentangled from an
alteration of α. Whether alone, or combined with a change in α, we
find it in all those many Gastropods whose whorls cannot all be
touched by the same enveloping cone, and whose spire is accordingly
described as concave or convex. The former condition, as we have it
in Cerithium, and in the cusp-like spire of Cassis,

Fig. 286. Trochus niloticus, L.
Dolium and some Cones, is much the commoner of the two. And such
tendency to decrease may lead to θ becoming a negative angle; in
which case we have a depressed spire, as in the Cypraeae.
When we find a “reversed shell,” a whelk or a snail for instance
whose spire winds to the left instead of to the right, we may describe
it math
e
mat
i
cally by the simple statement that the angle θ has
changed sign. In the genus Ampullaria, or Apple-snails, inhabiting
tropical or sub-tropical rivers, we have a remarkable condition; for in
certain “species” the spiral turns to the right, in others to the left, and
in others again we have a flattened {561} “discoid” shell; and
furthermore we have numerous intermediate stages, on either side,
shewing right and left-handed spirals of varying degrees of
acuteness527. In this case, the angle θ may be said to vary, within the
limits of a genus, from somewhere about 35° to somewhere about
125°.
The angle of retardation (β) is very small in Dentalium and Patella;
it is very large in Haliotis. It becomes infinite in Argonauta and in
Cypraea. Connected with the angle of retardation are the various

possibilities of contact or separation, in various degrees, between
adjacent whorls in the discoid, and between both adjacent and
opposite whorls in the turbinated shell. But with these phenomena we
have already dealt sufficiently.

Of Bivalve Shells.
Hitherto we have dealt only with univalve shells, and it is in these
that all the math
e
mat
i
cal problems connected with the spiral, or
helico-spiral, are best illustrated. But the case of the bivalve shell, of
Lamellibranchs or of Brachiopods, presents no essential difference,
save only that we have here to do with two conjugate spirals, whose
two axes have a definite relation to one another, and some freedom
of rotatory movement relatively to one another.
The generating curve is particularly well seen in the bivalve,
where it simply constitutes what we call “the outline of the shell.” It
is for the most part a plane curve, but not always; for there are
forms, such as Hippopus, Tridacna and many Cockles, or
Rhynchonella and Spirifer among the Brachiopods, in which the
edges of the two valves interlock, and others, such as Pholas, Mya,
etc., where in part they fail to meet. In such cases as these the
generating curves are conjugate, having a similar relation, but of
opposite sign, to a median plane of reference. A great variety of
form is exhibited by these generating curves among the bivalves. In
a good many cases the curve is ap
prox
i
mate
ly circular, as in Anomia,
Cyclas, Artemis, Isocardia; it is nearly semi-circular in Argiope. It is
ap
prox
i
mate
ly elliptical in Orthis and in Anodon; it may be called
semi-elliptical in Spirifer. It is a nearly rectilinear {562} triangle in
Lithocardium, and a curvilinear triangle in Mactra. Many apparently
diverse but more or less related forms may be shewn to be
deformations of a common type, by a simple application of the
math
e
mat
i
cal theory of “Transformations,” which we shall have to
study in a later chapter. In such a series as is furnished, for instance,
by Gervillea, Perna, Avicula, Modiola, Mytilus, etc., a “simple shear”
accounts for most, if not all, of the apparent differences.
Upon the surface of the bivalve shell we usually see with great
clearness the “lines of growth” which represent the successive
margins of the shell, or in other words the successive positions
assumed during growth by the growing generating curve; and we
have a good illustration, accordingly, of how it is char
ac
ter
is
tic of the

generating curve that it should constantly increase, while never
altering its geometric similarity.
Underlying these “lines of growth,” which are so char
ac
ter
is
tic of
a molluscan shell (and of not a few other organic formations), there
is, then, a “law of growth” which we may attempt to enquire into
and which may be illustrated in various ways. The simplest cases are
those in which we can study the lines of growth on a more or less
flattened shell, such as the one valve of an oyster, a Pecten or a
Tellina, or some such bivalve mollusc. Here around an origin, the so-
called “umbo” of the shell, we have a series of curves, sometimes
nearly circular, sometimes elliptical, and often asymmetrical; and
such curves are obviously not “concentric,” though we are often apt
to call them so, but are always “co-axial.” This manner of
arrangement may be illustrated by various analogies. We might for
instance compare it to a series of waves, radiating outwards from a
point, through a medium which offered a resistance increasing, with
the angle of divergence, according to some simple law. We may find
another, and perhaps a simpler illustration as follows:

Fig. 287.
In a very simple and beautiful
theorem, Galileo shewed that, if we
imagine a number of inclined planes,
or gutters, sloping downwards (in a
vertical plane) at various angles from
a common starting-point, and if we
imagine a number of balls rolling each
down its own gutter under the
influence of gravity (and without
hindrance from friction), then, at any
given instant, the locus of {563} all
these moving bodies is a circle passing
through the point of origin. For the
acceleration along any one of the sloping paths, for instance AB
(Fig. 287), is such that
AB = ½ g cos θ · t
2
= ½ g · AB ⁄ AC · t
2
.
Therefore
t
2
= 2 ⁄ g · AC.
That is to say, all the balls reach the circumference of the circle
at the same moment as the ball which drops vertically from A to C.
Where, then, as often happens, the generating curve of the shell
is ap
prox
i
mate
ly a circle passing through the point of origin, we may
consider the acceleration of growth along various radiants to be
governed by a simple math
e
mat
i
cal law, closely akin to that simple
law of acceleration which governs the movements of a falling body.
And, mutatis mutandis, a similar definite law underlies the cases
where the generating curve is continually elliptical, or where it
assumes some more complex, but still regular and constant form.
It is easy to extend the proposition to the particular case where
the lines of growth may be considered elliptical. In such a case we
have x
2
⁄ a
2
+ y
2
⁄ b
2
= 1, where a and b are the major and minor
axes of the ellipse.

Or, changing the origin to the vertex of the figure
x2
⁄ a
2
− 2x ⁄ a + y
2
⁄ b
2
= 0,
giving
(x − a)
2
⁄ a
2
+ y
2
⁄ b
2
= 1.
Then, transferring to polar coordinates, where r · cos θ = x,
r · sin θ = y, we have
(r · cos
2
θ) ⁄ a
2
− (2 cos θ) ⁄ a + (r · sin θ) ⁄ b
2
= 0,
{564}
which is equivalent to
r = 2 ab
2
cos θ ⁄ (b
2
cos
2
θ + a
2
sin
2
θ),
or, eliminating the sine-function,
r = 2 ab
2
cos θ ⁄ ((b
2
− a
2
) cos
2
θ + a
2
).
Obviously, in the case when a = b, this gives us the circular
system which we have already considered. For other values, or
ratios, of a and b, and for all values of θ, we can easily construct a
table, of which the following is a sample:

Fig. 288.
Chords of an ellipse, whose major and minor axes (a, b) are
in certain given ratios.
θ
a ⁄ b
= 1 ⁄ 3
1 ⁄ 2 2 ⁄ 3 1 ⁄ 1 3 ⁄ 2 2 ⁄ 1 3 ⁄ 1
0° 1·0 1·0 1·0 1·0 1·0 1·0 1·0
10 1·01 1·01 1·002 ·985 ·948 ·902 ·793
20 1·05 1·03 1·005 ·940 ·820 ·695 ·485
30 1·115 1·065 1·005 ·866 ·666 ·495 ·289
40 1·21 1·11 ·995 ·766 ·505 ·342 ·178
50 1·34 1·145 ·952 ·643 ·372 ·232 ·113
60 1·50 1·142 ·857 ·500 ·258 ·152 ·071
70 1·59 1·015 ·670 ·342 ·163 ·092 ·042
80 1·235 ·635 ·375 ·174 ·078 ·045 ·020
90 0·0 0·0 0·0 0·0 0·0 0·0 0·0
The coaxial ellipses which we then draw, from
the values given in the table, are such as are
shewn in Fig. 288 for the ratio a ⁄ b = 3 ⁄ 1, and in
Fig. 289 for the ratio a ⁄ b = 1 ⁄ 2 ; these are fair
ap
prox
i
ma
tions to the actual out
lines, and to the
actual ar
range
ment of the lines of growth, in
such forms as Sole
cur
tus or Cul
tel
lus, and in Tel
‐
lina or Psam
mobia. It is not dif
ficult to intro
duce
a cons
tant into our equa
tion to meet the case of
a shell which is somewhat unsymmetrical on
either side of the median axis. It is a somewhat
more troublesome matter, however, to bring
these con
fi
gur
a
tions into relation with a “law of
growth,” as was so easily done in the case of the
circular figure: in other words, to {565} formulate a law of acceleration
according to which points starting from the origin O, and moving

along radial lines, would all lie, at any future epoch, on an ellipse
passing through O; and this calculation we need not enter into.
Fig. 289.
All that we are immediately concerned with is the simple fact that
where a velocity, such as our rate of growth, varies with its direction,
—varies that is to say as a function of the angular divergence from a
certain axis,—then, in a certain simple case, we get lines of growth
laid down as a system of coaxial circles, and, when the function is a
more complex one, as a system of ellipses or of other more
complicated coaxial figures, which figures may or may not be
symmetrical on either side of the axis. Among our bivalve mollusca
we shall find the lines of growth to be ap
prox
i
mate
ly circular in, for
instance, Anomia; in Lima (e.g. L. subauriculata) we have a system
of nearly symmetrical ellipses with the vertical axis about twice the
transverse; in Solen pellucidus, we have again a system of lines of
growth which are not far from being symmetrical ellipses, in which
however the transverse is between three and four times as great as
the vertical axis. In the great majority of cases, we have a similar
phenomenon with the further complication of slight, but occasionally
very considerable, lateral asymmetry.

In certain little Crustacea (of the genus Estheria) the carapace
takes the form of a bivalve shell, closely simulating that of a {566}
lamellibranchiate mollusc, and bearing lines of growth in all respects
analogous to or even identical with those of the latter. The
explanation is very curious and interesting. In ordinary Crustacea the
carapace, like the rest of the chitinised and calcified integument, is
shed off in successive moults, and is restored again as a whole. But
in Estheria (and one or two other small crustacea) the moult is
incomplete: the old carapace is retained, and the new, growing up
underneath it, adheres to it like a lining, and projects beyond its
edge: so that in course of time the margins of successive old
carapaces appear as “lines of growth” upon the surface of the shell.
In this mode of formation, then (but not in the usual one), we obtain
a structure which “is partly old and partly new,” and whose
successive increments are all similar, similarly situated, and enlarged
in a continued progression. We have, in short, all the conditions
appropriate and necessary for the development of a logarithmic
spiral; and this logarithmic spiral (though it is one of small angle)
gives its own character to the structure, and causes the little
carapace to partake of the char
ac
ter
is
tic conformation of the
molluscan shell.
The essential simplicity, as well as the great regularity of the
“curves of growth” which result in the familiar con
fi
gur
a
tions of our
bivalve shells, sufficiently explain, in a general way, the ease with
which they may be imitated, as for instance in the so-called “artificial
shells” which Kappers has produced from the conchoidal form and
lamination of lumps of melted and quickly cooled paraffin528.
In the above account of the math
e
mat
i
cal form of the bivalve shell, we have supposed,
for simplicity’s sake, that the pole or origin of the system is at a point where all the
successive curves touch one another. But such an arrangement is neither theoretically
probable, nor is it actually the case; for it would mean that in a certain direction growth fell,
not merely to a minimum, but to zero. As a matter of fact, the centre of the system (the
“umbo” of the conchologists) lies not at the edge of the system, but very near to it; in other
words, there is a certain amount of growth all round. But to take account of this condition

would involve more troublesome mathematics, and it is obvious that the foregoing
illustrations are a sufficiently near approximation to the actual case. {567}
Among the bivalves the spiral angle (α) is very small in the
flattened shells, such as Orthis, Lingula or Anomia. It is larger, as a
rule, in the Lamellibranchs than in the Brachiopods, but in the latter
it is of considerable magnitude among the Pentameri. Among the
Lamellibranchs it is largest in such forms as Isocardia and Diceras,
and in the very curious genus Caprinella; in all of these last-named
genera its magnitude leads to the production of a spiral shell of
several whorls, precisely as in the univalves. The angle is usually
equal, but of opposite sign, in the two valves of the Lamellibranch,
and usually of opposite sign but unequal in the two valves of the
Brachiopod. It is very unequal in many Ostreidae, and especially in
such forms as Gryphaea, or in Caprinella, which is a kind of
exaggerated Gryphaea. Occasionally it is of the same sign in both
valves (that is to say, both valves curve the same way) as we see
sometimes in Anomia, and much better in Productus or
Strophomena.

Fig. 290. Caprinella
adversa. (After Woodward.)
Fig. 291. Section of Productus
(Strophomena) sp. (From Woods.)
Owing to the large growth-factor of the generating curve, and
the comparatively small angle of the spiral, the whole shell seldom
assumes a spiral form so conspicuous as to manifest in a typical way
the helical twist or shear which is so conspicuous in the {568} majority
of univalves, or to let us measure or estimate the magnitude of the
apical angle (θ) of the enveloping cone. This however we can do in
forms like Isocardia and Diceras; while in Caprinella we see that the
whorls lie in a plane perpendicular to the axis, forming a discoidal
spire. As in the latter shell, so also universally among the
Brachiopods, there is no lateral asymmetry in the plane of the
generating curve such as to lead to the development of a helix; but
in the majority of the Lamellibranchiata it is obvious, from the
obliquity of the lines of growth, that the angle θ is significant in
amount.
The so-called “spiral arms” of Spirifer and many other
Brachiopods are not difficult to explain. They begin as a single

Fig. 292. Skeletal loop of
Terebratula. (From Woods.)
structure, in the form
of a loop of shelly substance, attached
to the dorsal valve of the shell, in the
neighbourhood of the hinge. This loop
has a curvature of its own, similar to
but not necessarily identical with that
of the valve to which it is attached;
and this curvature will tend to be
developed, by continuous and
symmetrical growth, into a fully
formed logarithmic spiral, so far as it is
permitted to do so under the
constraint of the shell in which it is
contained. In various Terebratulae we
see the spiral growth of the loop, more or less flattened and
distorted by the restraining pressure of the ventral valve. In a
number of cases the loop remains small, but gives off two nearly
parallel branches or offshoots, which continue to grow. And these,
starting with just such a slight curvature as the loop itself possessed,
grow on and on till they may form close-wound spirals, always
provided that the “spiral angle” of the curve is such that the
resulting spire can be freely contained within the cavity of the shell.
Owing to the bilateral symmetry of the whole system, the case will
be rare, and unlikely to occur, in which each separate arm will coil
strictly in a plane, so as to constitute a discoid spiral; for the original
{569} direction of each of the two branches, parallel to the valve (or
nearly so) and outwards from the middle line, will tend to constitute
a curve of double curvature, and so, on further growth, to develop
into a helicoid. This is what actually occurs, in the great majority of
cases. But the curvature may be such that the helicoid grows
outwards from the middle line, or inwards towards the middle line, a
very slight difference in the initial curvature being sufficient to direct
the spire the one way or the other; the middle course of an
undeviating discoid spire will be rare, from the usual lack of any
obvious controlling force to prevent its deviation. The cases in which

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