An illustration showing how to construct a center and radius of a circle that will tangent the three sides of a triangle. "Bisect two of the angles in the triangle, and the crossing C is the center of the required circle."

Construction Of The Center And Radius Of A Circle Tangent To Triangle Sides

An illustration showing how to construct a center and radius of a circle that will tangent the three…

"Three center arches, employed in French Flamboyant." — The Encyclopedia Britannica, 1910

Center Arch

"Three center arches, employed in French Flamboyant." — The Encyclopedia Britannica, 1910

"Four center arches, employed in the Perpendicular and Tudor periods." — The Encyclopedia Britannica, 1910

Center Arch

"Four center arches, employed in the Perpendicular and Tudor periods." — The Encyclopedia Britannica,…

"Why does a person carrying a weight upon his back stoop forward? In order to bring the center of gravity of his body and the load over his feet. If held in this position, he would fall backwards, as the direction of the center of gravity would fall beyond his heels." — Wells, 1857

Center of Gravity

"Why does a person carrying a weight upon his back stoop forward? In order to bring the center of gravity…

"Why does a person carrying a weight upon his back stoop forward? In order to bring the center of gravity of his body and the load over his feet, he assumes this position." — Wells, 1857

Center of Gravity

"Why does a person carrying a weight upon his back stoop forward? In order to bring the center of gravity…

"The center of gravity, in any body or system of bodies is that point upon which the body, or system of bodies, acted upn only by gravity will balance itself in all positions." -Comstock 1850

Center of Gravity

"The center of gravity, in any body or system of bodies is that point upon which the body, or system…

"But if one of the balls be heavier than the other, then the centre of gravity will, in proportion, approach the larger ball." -Comstock 1850

Center of Gravity

"But if one of the balls be heavier than the other, then the centre of gravity will, in proportion,…

"In a body of equal thickness, as a board, or a slab of marble, but otherwise of an irregular shape, the centre of gravity may be found by suspending it, first from one point, and then from another, and marking, by means of a plumb line, the perpendicular ranges from the point of suspension. the centre of gravity will be the point where these two lines cross each other." -Comstock 1850

Center of Gravity

"In a body of equal thickness, as a board, or a slab of marble, but otherwise of an irregular shape,…

"If w and W, fig. 3, be two bodies of known weight, their center of gravity will be at C." —Hallock 1905

Center of Gravity

"If w and W, fig. 3, be two bodies of known weight, their center of gravity will be at C." —Hallock…

"The part of the body in which the centre of gravity is situated, may be found, in some cases, by balancing it on a point. Thus the centre of gravity of the poker represented [here] lies directly over the point on which it is balanced." —Quackenbos 1859

Center of Gravity

"The part of the body in which the centre of gravity is situated, may be found, in some cases, by balancing…

"When such a surface is irregular in shape, suspend it at any point, so that it may move freely, and when it has come to rest, drop a plumb line from the point of suspension and mark its direction on the surface. Do the same at any other point, and the centre of gravity will lie where the two line intersect." —Quackenbos 1859

Center of Gravity

"When such a surface is irregular in shape, suspend it at any point, so that it may move freely, and…

"Where five blocks are placed in this position, the point of gravity is near the centre of the thrd block, and is within the base as shown by the plumb line. But on adding another block, the gravitation point falls beyond the base, and the whole will now fall by its own weight." -Comstock 1850

Center of Gravity of Standing Blocks

"Where five blocks are placed in this position, the point of gravity is near the centre of the thrd…

"Find the center of gravity of two of the bodies, as W1, and W4, in fig 4. Assume that the weight of both bodies is concentrated at C1, and find the center of gravity of this combined weight at C1, and the weight of W2; let it be at C2; then find the center of gravity of the combined weights of W1, W4, W2 (concentrated at C2), and W2; let it be at C; then C will be the center of gravity of the four bodies." —Hallock 1905

Compound Center of Gravity

"Find the center of gravity of two of the bodies, as W1, and W4, in fig 4. Assume that the weight of…

"A load of hay...upsets where one wheel rises by little above the other, because it is broader on the top than the distance of the wheels from each other; while a load of stone is very rarely turned over..." -Comstock 1850

Center of Gravity of a Load of Hay

"A load of hay...upsets where one wheel rises by little above the other, because it is broader on the…

"To find the center of gravity of any irregular plane figure, but of uniform thickness throughout, divide one of the parallel surfaces into triangles, parallelograms, circles, ellipses, etc., according to the shape of the figure; find the area and center of gravity of each part separately, and combine the center of gravity thus found in the same manner." —Hallock 1905

Center of Gravity of an Irregular Plane

"To find the center of gravity of any irregular plane figure, but of uniform thickness throughout, divide…

"When a line of direction falls within the base, a body stands when not, it falls... On the same principle, a load of stone may pass safely over a hillside, on which a load of hay would be overturned [as shown by the line of direction in this illustration]." —Quackenbos 1859

Line of Direction from the Center of Gravity of an Object

"When a line of direction falls within the base, a body stands when not, it falls... On the same principle,…

"When two bodies of equal weight are connected by a rod, the centre of gravity will lie in the middle of that rod. When two bodies of unequal weight are so connected, the centre of gravity will be nearer to the heavier one. These principles are illustrated [here], in which C represents the centre of gravity." —Quackenbos 1859

Center of Gravity of Rod with Weights

"When two bodies of equal weight are connected by a rod, the centre of gravity will lie in the middle…

"When a ball is rolling on a horizontal plane, the centre of gravity is not raised, but moves in a straight line, parallel to the surface of the plane on which it rolls, and is consequently always directly over its centre of gravity." -Comstock 1850

Center of Gravity of a Rolling Ball

"When a ball is rolling on a horizontal plane, the centre of gravity is not raised, but moves in a straight…

"In a body free to move, the center of gravity will lie in a vertical plumb-line drawn through the point of support. Therefore, to find the position of the center of gravity of an irregular solid, as the crank, Fig 8, suspended it at some point, as B, so that it will move freely. Drop a plumb line from the point of suspension and mark its direction. Suspend the body at another point, as A, and repeat the process. The intersection C of the two lines will be directly over the center of gravity." —Hallock 1905

Center of Gravity of a Solid

"In a body free to move, the center of gravity will lie in a vertical plumb-line drawn through the point…

"But, suppose the same bar or iron, whose inertia was overcome by raising the centre, to have balls of different weights attached to its ends; then the centre of inertia would no longer remain in the middle of the bar, but would be changed to the point A..." -Comstock 1850

Center of Inertia

"But, suppose the same bar or iron, whose inertia was overcome by raising the centre, to have balls…

"A body's center of mass is the point about which all matter composing the body may be balanced. It is also called the center of inertia. In some cases, this is also the center of gravity." -Avery 1895

Center of Mass

"A body's center of mass is the point about which all matter composing the body may be balanced. It…

"Let any irregularly shpaed body, as a stone or chair, be suspended so as to move freely. Drop a plumb line from the point of the suspendsion, and make it fast or mark its direction. The center of mass will lie in this line. From a second point, not in the line already determined, suspend the body; let it fall a plumb line as before. The center of mass will lie in this line also. But to lie in both lines, it must lie at their intersection." -Avery 1895

Finding the Center of Mass

"Let any irregularly shpaed body, as a stone or chair, be suspended so as to move freely. Drop a plumb…

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through its center, and a similar point is found on the line at an equal distance beyond the center." — Ford, 1912

Symmetry center

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through…

Circle with 36 degree angles marked. This diagram can be used with the following trig problem: Locate the centers of the holes B and C by finding the distance each is to the right and above the center O. The radius of the circle is 1.5 inches. Compute correct to three decimal places.

Circle With 36 degree Angles and Radius 1.5 in.

Circle with 36 degree angles marked. This diagram can be used with the following trig problem: Locate…

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height of the point P above the surface; it is required to find the distance from the point P to the horizon at A.

Circle With Center o and Radius r with point P

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height…

Illustration used to find the center of a circle.

Center of a Circle

Illustration used to find the center of a circle.

Circle with chord AB=2 ft. and radius OA = 3 ft.. Triangle AOC is a right triangle. Angle AOC=half angle AOB, and the central angle AOB has the same measure as the arc AnB.

Circle With a Chord of 2 ft. and a Radius of 3 ft.

Circle with chord AB=2 ft. and radius OA = 3 ft.. Triangle AOC is a right triangle. Angle AOC=half angle…

Illustration of a square, with diagonals drawn, circumscribed about a circle. This can also be described as a circle inscribed in a square. The diagonals of the square intersect at the center of both the square and the circle. The diagonals coincide with the diameter of the circle.

Square Circumscribed About A Circle

Illustration of a square, with diagonals drawn, circumscribed about a circle. This can also be described…

Illustration of a square, with 1 diagonals drawn, circumscribed about a circle. This can also be described as a circle inscribed in a square. The diagonal goes through the center of both the square and the circle and coincides with the diameter of the circle.

Square Circumscribed About A Circle

Illustration of a square, with 1 diagonals drawn, circumscribed about a circle. This can also be described…

Illustration of a square circumscribed about a circle. This can also be described as a circle inscribed in a square.

Square Circumscribed About A Circle

Illustration of a square circumscribed about a circle. This can also be described as a circle inscribed…

Illustration of a square, with diagonals drawn, inscribed in a circle. This can also be described as a circle circumscribed about a square. The diagonals, which are also the diameter of the circle, intersect at the center of both the square and the circle.

Square Inscribed In A Circle

Illustration of a square, with diagonals drawn, inscribed in a circle. This can also be described as…

Illustration used to prove "If two circumferences meet at a point which is not on their line of centers, they also meet in one other point."

Circumferences of 2 Circles

Illustration used to prove "If two circumferences meet at a point which is not on their line of centers,…

A market cross is a structure used to mark a market square in market towns, originally from Western European architecture. Market crosses can be found in most market towns in Britain, with those in Scotland known as "mercat crosses". British emigrants often installed such crosses in their new cities and several can be found in Canada and Australia.

Market Cross

A market cross is a structure used to mark a market square in market towns, originally from Western…

Illustration of blueprint used by highway engineers to widen the pavement on the inside of the curve of a road.

Curve in Pavement of Road

Illustration of blueprint used by highway engineers to widen the pavement on the inside of the curve…

"If we suppose a spectator placed at G, in the Earth's center, he would see the moon E, among the stars at I, whereas without changing the position of the moon, if that body is seen from A, on the surface of the Earth, it would appear among the stars at K. Now I is the true and K the apparent place of the moon, the space between them, being the Moon's parallax." —Comstock, 1850

Diurnal Parallax

"If we suppose a spectator placed at G, in the Earth's center, he would see the moon E, among the stars…

"The Earth, whose diameter is 7,912 miles, is represented by the globe, or sphere. The straight line passing through its center, and about which it turns, is called its axis, and the two extremities of the axis are the poles of the Earth, A being the north pole, and B the south pole. The line C D, crossing the axis, passes quite round the Earth, and divides it into two equal parts. This is called the equinoctial line, or the equator. That part of the Earth situated north of this line, is caled the northern hemisphere, and that part south of it, the southern hemisphere. The small circles E F and G H, surrounding or including the poles, are called the polar circles." —Comstock, 1850

Earth Divisions

"The Earth, whose diameter is 7,912 miles, is represented by the globe, or sphere. The straight line…

Illustration of half of an ellipse. "The ordinates of two corresponding points in an ellipse and its auxiliary circle are in the ratio b:a."

Corresponding Points in an Ellipse and Circle

Illustration of half of an ellipse. "The ordinates of two corresponding points in an ellipse and its…

Illustration showing that if one diameter is conjugate to a second, the second is conjugate to the first.

Conjugate Diameters of an Ellipse

Illustration showing that if one diameter is conjugate to a second, the second is conjugate to the first.

Illustration of half of an ellipse and its auxiliary circle used to construct an ellipse by points, having given its two axes.

Construction of an Ellipse

Illustration of half of an ellipse and its auxiliary circle used to construct an ellipse by points,…

Illustration of half of an ellipse. "If d denotes the abscissa of a point of an ellipse, r and r' its focal radii, then r'=a+ed, r=a-ed."

Focal Radii of an Ellipse

Illustration of half of an ellipse. "If d denotes the abscissa of a point of an ellipse, r and r' its…

A graph of tan ellipse with Foci and Center.

Graph of Ellipse with Foci and Center Labeled

A graph of tan ellipse with Foci and Center.

Illustration of an ellipse with foci F' and F, major axis A' to A, minor axis B' to B, and center O.

Ellipse With Parts Labeled

Illustration of an ellipse with foci F' and F, major axis A' to A, minor axis B' to B, and center O.

Illustration of half of an ellipse. "If through a point P of an ellipse a line is drawn bisecting the angle between one of the focal radii and the other produced, every point in this line except P is without the curve."

Line Bisecting Angle Between Focal Radii on Ellipse

Illustration of half of an ellipse. "If through a point P of an ellipse a line is drawn bisecting the…

Illustration of half of an ellipse. The square of the ordinate of a point in an ellipse is to the product of the segments of the major axis made by the ordinate as the square of b to the square of a.

Ordinate and Major Axis of Ellipse

Illustration of half of an ellipse. The square of the ordinate of a point in an ellipse is to the product…

Illustration showing that tangents drawn at the ends of any diameter are parallel to each other.

Parallel Tangents to an Ellipse

Illustration showing that tangents drawn at the ends of any diameter are parallel to each other.

Diagram of an ellipse that can used to illustrate the different parts. Segment MN is the major axis, segment CD is the conjugate (minor) axis, and point O is the center of the ellipse. Both foci are also labeled in the illustration.

Parts of Ellipse

Diagram of an ellipse that can used to illustrate the different parts. Segment MN is the major axis,…

Illustration of half of an ellipse. "The sum of the distances of any point from the foci of an ellipse is greater than or less than 2a, according as the point is without or within the curve."

Point Distances to Foci on Ellipse

Illustration of half of an ellipse. "The sum of the distances of any point from the foci of an ellipse…

Illustration of how to draw a tangent to an ellipse from an external point.

Tangent From External Point to an Ellipse

Illustration of how to draw a tangent to an ellipse from an external point.

Illustration showing the tangents drawn at two corresponding points of an ellipse and its auxiliary circle cut the major axis produced at the same point.

Tangents to an Ellipse

Illustration showing the tangents drawn at two corresponding points of an ellipse and its auxiliary…

An illustration showing how to construct an equilateral triangle inscribed in a circle. "With the radius of the circle and center C draw the arc DFE; with the same radius, and D and E as centers, set off the points A and B. Join A and B, B and C, C and A, which will be the required triangle."

Construction Of An Equilateral Triangle Inscribed In A Circle

An illustration showing how to construct an equilateral triangle inscribed in a circle. "With the radius…

"FESS POINT. The exact centre of the escutcheon, as seen in the annexed example." -Hall, 1862

Fess Point

"FESS POINT. The exact centre of the escutcheon, as seen in the annexed example." -Hall, 1862

An illustration showing how to construct a hexagon in a given circle. "The radius of the circle is equal to the side of the hexagon."

Construction Of A Hexagon In A Circle

An illustration showing how to construct a hexagon in a given circle. "The radius of the circle is equal…

"HONOUR POINT. That part of the shield between the precise middle chief and the fess point. In the annexed example the large dot in the centre shows the fess point; the point within the letter D, the honour point." -Hall, 1862

Honour Point

"HONOUR POINT. That part of the shield between the precise middle chief and the fess point. In the annexed…

An illustration of a decorative doodad with a horn and doe in center.

Horn with Doe in Center

An illustration of a decorative doodad with a horn and doe in center.

The central part of a car wheel (or fan or propeller etc) through which the shaft or axle passes

Wheel Hub

The central part of a car wheel (or fan or propeller etc) through which the shaft or axle passes

"When the rays diverge from a point beyond the center of curvature, as B, the focus falls on the same axis, at a distrance from the mirror greater than that of the principal focus, and less than that of the center of curvature." -Avery 1895

Rays Diverging from Beyond the Center of Curvature on a Concave Mirror

"When the rays diverge from a point beyond the center of curvature, as B, the focus falls on the same…

An illustration showing how to construct a pentagon inscribed in a circle. "Draw the diameter AB, and from the center C erect the perpendicular CD. Bisect the radius AC at E; with E as center, and DE as radius, draw the arc DE, and the straight line DF is the length of the side of the pentagon."

Construction Of A Pentagon Inscribed In A Circle

An illustration showing how to construct a pentagon inscribed in a circle. "Draw the diameter AB, and…

An illustration showing how to construct a pentagon on a given line. "From B erect BC perpendicular to and half the length of AB; join A and C prolonged to D; with C as center and CB as radius, draw the arc BD; then the chord BB is the radius of the circle circumscribing the pentagon. With A and B as centers, and BD as radius, draw the cross O in the center."

Construction Of A Pentagon On A Line

An illustration showing how to construct a pentagon on a given line. "From B erect BC perpendicular…

Given any three circles, the common chords meet at one point.

Radical Center of 3 Circles

Given any three circles, the common chords meet at one point.

A small mountain town with a Church in the center.

Small Mountain Town

A small mountain town with a Church in the center.