During new and full moon, the earth, moon, and sun are all in the same straight line, but, that during the first and last quarters, they are at right angles. The portions of the earth and moon turned toward the sun are illumined, the shaded portions are in the darkness. To an observer on the earth, the moon, at <em>a</em>, appears new, since the dark part is turned toward the person; at <em>b</em>, however, it must appear full, since the illumined portions are toward the person. At <em>c</em>, and <em>d</em>, the positions of the quarters, only one-half of the illumined half, or one quarter, is seen.

Cause of the Phases of the Moon

During new and full moon, the earth, moon, and sun are all in the same straight line, but, that during…

Illustration showing a centroid, "the curve passing through the successive positions of the instantaneous centre of a body having a combined motion of rotation and translation is called a centroid." A smooth curve passes through the successive positions of the instantaneous centers (all the centers marked o)will be the centroid ab.

Centroid

Illustration showing a centroid, "the curve passing through the successive positions of the instantaneous…

Possible forms of the invariable cone by means of the intersections with a concentric spherical surface.

Invariable Cone

Possible forms of the invariable cone by means of the intersections with a concentric spherical surface.

"Since these curves are not closed, one pair cannot be used for continuous motion; but a pair of such curves may be well adapted to sectional wheels requiring a varying angular velocity." This figure shows an example of the process.

Motion of Open Curves

"Since these curves are not closed, one pair cannot be used for continuous motion; but a pair of such…

One Domino

Domino

One Domino

"If two equal ellipses, each turning about one of its foci, are placed in contact in such a way that the distance between the axes O<SUB>1</SUB> and O<SUB>2</SUB> is equal to the major axis of the ellipses, we shall find that they will be in contact on the line of centres and that the rolling arcs are of equal length."

Rolling of Equal Ellipses

"If two equal ellipses, each turning about one of its foci, are placed in contact in such a way that…

Illustration of the rolling of equal hyperbolas. If two equal hyperbolas are placed so that the distances between their foci O<SUB>1</SUB> and O<SUB>2</SUB>, and d and e, are each equal to fg=hk, they will make contact at some point c.

Rolling of Equal Hyperbolas

Illustration of the rolling of equal hyperbolas. If two equal hyperbolas are placed so that the distances…

Illustration of the rolling of equal parabolas. "The two parabolas may be considered as two ellipses with one focus of each removed to infinity."

Rolling of Equal Parabolas

Illustration of the rolling of equal parabolas. "The two parabolas may be considered as two ellipses…

"The Gyroscope is an instrument constructed by M. Foucault to make the rotation of the earth visible. The principle on which it proceeds is this&ndash; that, unless gravity intervene, a rotating body will not alter the direction in which its permanent axis points. In the gyroscope there is a rotating metallic disk, the middle point of whose axis is also the center of gravity of the machine. By this device the action of gravity is eliminated."&mdash;(Charles Leonard-Stuart, 1911)

Gyroscope

"The Gyroscope is an instrument constructed by M. Foucault to make the rotation of the earth visible.…

"Foucault's Gyroscope. GYROSCOPE. An instrument consisting of a fly-wheel, the axis of which can turn freely in any direction, designed to illustrate the dynamics of rotating bodies." -Whitney, 1911

Foucault's Gyroscope

"Foucault's Gyroscope. GYROSCOPE. An instrument consisting of a fly-wheel, the axis of which can turn…

"Gyrostat. An instrument for illustrating the dynamics of rotation, composed of a box or case having a sharp bearing-edge in the form of a regular polygon, and containing a fly-wheel having its center and its direction of rotation in the plane of the bearing-edge." -Whitney, 1911

Gyrostat

"Gyrostat. An instrument for illustrating the dynamics of rotation, composed of a box or case having…

Illustration showing how a rolling logarithmic spiral can be constructed by taking successive values of theta, starting from oc. The values of r can be calculated and then the curve can be plotted.

Construction of Logarithmic Spiral

Illustration showing how a rolling logarithmic spiral can be constructed by taking successive values…

Illustration showing the rolling of two logarithmic spirals of equal obliquity.

Rolling of Logarithmic Spirals

Illustration showing the rolling of two logarithmic spirals of equal obliquity.

If the earth were uniformly covered with a layer of water, the passage of the moon over any place as at <em>a</em>, would cause the water to lose its globular form, become bulged at <em>a</em>, and <em>b</em>, and flattened at <em>c</em>, and <em>d</em>. In other words, the water would become <em>deeper</em> at <em>a</em>, and <em>b</em>, at the parts of the earth nearest and farthest from the moon and <em>shallower</em> in all places 90 degrees or at right angles to these points.

Lunar Tide

If the earth were uniformly covered with a layer of water, the passage of the moon over any place as…

"Let S be the Sun, E the Earth, and A, B, C, D, F, the Moon in different parts of her orbit. Now when the Moon changes, or is in conjunction with the Sun, as at A, her dark side is turned towards the Earth, and she is invisible, as represented at a. The Sun always shines on one half of the Moon, in every direction, as represented at A and B, on the inner circle; but we at the Earth can see only such portions of the enlightened part as are turned towards us. After her change, when she has moved from A to B, a small part of her illuminated side comes in sight, and she appears horned, as at b, and is then called the new Moon. When she arrives at C, severel days afterwards, one half of her disc is visible, and she appears as at c, her appearance being the same in both circles. At this point she is said to be in her first quarter, because she has passed through a quarter of her orbit, and is 90 degrees from the place of her conjunction with the Sun. At D, she shows us still more of her enlightened side, and is then said to appear gibbous as at d. When she comes to F, her whole enlightened side is turned towards the Earth, and she appears in all the spendor of a full Moon." &mdash;Comstock, 1850

Moon Phases

"Let S be the Sun, E the Earth, and A, B, C, D, F, the Moon in different parts of her orbit. Now when…

Illustration showing that the rolling of non-cylindrical surfaces. "If the angular velocity ratio of two rolling bodies is not a constant, the pitch lines take, the conditions of pure rolling contact should be fulfilled, namely, the point of contact must be on the line of centres, and the rolling arcs must be of equal length.

Rolling of Non-cylindrical Surfaces

Illustration showing that the rolling of non-cylindrical surfaces. "If the angular velocity ratio of…

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 1

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 10

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 100

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 101

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 11

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 12

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 13

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 14

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 15

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 16

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 17

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 18

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 19

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 2

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 20

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 21

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 22

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 23

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 24

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 25

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 26

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 27

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 28

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 29

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 3

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 30

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 31

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 32

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 33

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 34

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 35

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 36

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 37

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 38

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 39

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 4

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 40

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 41

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 42

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 43

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 44

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 45

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 46

Geometric pattern for translation and rotation exercises.

Geometric pattern for translation and rotation exercises.

Geometric Block Pattern 47

Geometric pattern for translation and rotation exercises.