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A sphere with arcs drawn.

Sphere With Arcs

A sphere with arcs drawn.

A sphere with arcs and poles. "The distance of all points in the circumference of a circle of a sphere from its poles are equal."

Sphere With Arcs and Poles

A sphere with arcs and poles. "The distance of all points in the circumference of a circle of a sphere…

A sphere with quadrants drawn on it.

Sphere With Quadrants

A sphere with quadrants drawn on it.

A sphere with quadrants and poles. "A point on the surface of a sphere, which is at the distance of a quadrant from each of two other points, not the extremities of a diameter, is a pole of the great circle passing through theses points."

Sphere With Quadrants and Poles

A sphere with quadrants and poles. "A point on the surface of a sphere, which is at the distance of…

Illustration of a material sphere that can be used to find its diameter.

Material Sphere

Illustration of a material sphere that can be used to find its diameter.

Illustration of a sphere with a tangent plane. "A plane perpendicular to a radius at its extremity is tangent to the sphere."

Sphere With Tangent Plane

Illustration of a sphere with a tangent plane. "A plane perpendicular to a radius at its extremity is…

Illustration of a sphere with a tangent plane. "A plane perpendicular to a radius at its extremity is tangent to the sphere."

Sphere With Tangent Plane

Illustration of a sphere with a tangent plane. "A plane perpendicular to a radius at its extremity is…

Illustration of a sphere inscribed in a tetrahedron.

Sphere Inscribed in Tetrahedron

Illustration of a sphere inscribed in a tetrahedron.

Illustration of a sphere inscribed in a tetrahedron.

Sphere Inscribed in Tetrahedron

Illustration of a sphere inscribed in a tetrahedron.

Illustration of two intersecting spheres. "The intersection of two spherical surfaces is the circumference of a circle whose plane is perpendicular to the line joining the centres of the surfaces and whose centre is in that line."

Intersecting Spheres

Illustration of two intersecting spheres. "The intersection of two spherical surfaces is the circumference…

Illustration of a sphere with arcs drawn.

Arcs Drawn on Sphere

Illustration of a sphere with arcs drawn.

Illustration of a sphere with a polyhedral angle whose vertex is at the center of the sphere and whose face angles are measured by the sides of the polygon, whose dihedral angles have the same numerical measure as the angles of the polygon.

Polyhedral Angle in a Sphere

Illustration of a sphere with a polyhedral angle whose vertex is at the center of the sphere and whose…

Illustration of a sphere with a polyhedral angle.

Polyhedral Angle in a Sphere

Illustration of a sphere with a polyhedral angle.

Illustration of polar and spherical triangles.

Polar and Spherical Triangles

Illustration of polar and spherical triangles.

Illustration of polar triangles.

Polar Triangles in Sphere

Illustration of polar triangles.

Illustration of symmetrical spherical triangles.

Symmetric Spherical Triangles

Illustration of symmetrical spherical triangles.

Illustration of symmetrical spherical triangles.

Symmetric Spherical Triangles

Illustration of symmetrical spherical triangles.

Illustration of symmetrical spherical triangles.

Symmetric Spherical Triangles

Illustration of symmetrical spherical triangles.

Illustration of equal spheres with equal triangles.

Equal Spheres With Triangles

Illustration of equal spheres with equal triangles.

Illustration of equal spheres with equilateral and equiangular triangles.

Equal Spheres With Triangles

Illustration of equal spheres with equilateral and equiangular triangles.

Illustration of an isosceles spherical triangle.

Isosceles Spherical Triangle

Illustration of an isosceles spherical triangle.

Illustration of an isosceles spherical triangle.

Isosceles Spherical Triangle

Illustration of an isosceles spherical triangle.

Illustration of a semicircle with chords and radii.

Semicircle With Chords and Radii

Illustration of a semicircle with chords and radii.

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…

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 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. 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 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 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 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…

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.

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 a hyperbola and its auxiliary circle. "Any ordinate of a hyperbola is to the tangent from its foot to the auxiliary circle as b is to a."

Auxiliary Circle and Hyperbola

Illustration of a hyperbola and its auxiliary circle. "Any ordinate of a hyperbola is to the tangent…

Illustration of a circle with the radius OA drawn and labeled.

Radius of Circle

Illustration of a circle with the radius OA drawn and labeled.

Illustration of a circle with the arc aeb drawn and labeled.

Arc of Circle

Illustration of a circle with the arc aeb drawn and labeled.

Illustration of a circle with the chord eb drawn and labeled.

Chord of Circle

Illustration of a circle with the chord eb drawn and labeled.

Illustration of a circle with a sector. "A sector of a circle is the space included between an arc and two radii drawn to the extremities of the arc."

Sector of Circle

Illustration of a circle with a sector. "A sector of a circle is the space included between an arc and…

Illustration of a circle with two intersecting diameters.

Diameters of a Circle

Illustration of a circle with two intersecting diameters.

Illustration of a circle with 6 equal sectors.

Sectors of Circle

Illustration of a circle with 6 equal sectors.

Illustration of a circle with an inscribed angle.

Inscribed Angle in a Circle

Illustration of a circle with an inscribed angle.

Illustration of intersecting circles. "If two circles intersect each other, the line joining their centers bisects at right angles the line joining the two points of intersection."

Intersecting Circles

Illustration of intersecting circles. "If two circles intersect each other, the line joining their centers…

Illustration of tangent circles. One circle is said to be tangent to another circle when they touch each other at one point only.

Tangent Circles

Illustration of tangent circles. One circle is said to be tangent to another circle when they touch…

"If from any point on the circumference of a circle, a perpendicular be let fall upon a given diameter, this perpendicular will be a mean proportional between the two parts into which it divides the diameter."

Circle With a Perpendicular Drawn to the Diameter

"If from any point on the circumference of a circle, a perpendicular be let fall upon a given diameter,…

"If AOB is an angle of 1 ° on the larger circle, it is also 1 ° on the smaller concentric circle, and the length of the arc AB is to the length of the arc CD as to radius OB is to the radius OD; or, arc AB:arc CD = OB:OD."

Concentric Circles With Angle of 1 °

"If AOB is an angle of 1 ° on the larger circle, it is also 1 ° on the smaller concentric circle,…

"In trigonometry, the arcs of circles are used to measure angles. All angles are supposed to have their vertexes at the center O of the circle, one side of the angle lying to the right of O, and coinciding with the horizontal diameter, as OB."

Arcs and Angles of a Trigonometric Circle

"In trigonometry, the arcs of circles are used to measure angles. All angles are supposed to have their…

Bones of the arm and hand. Labels: a, large end of ulna; b, radius; c, small end of the ulna; d, carpal bones; e, metacarpal bones; f, first row of the bones of the fingers.

Bones of the Arm and Hand

Bones of the arm and hand. Labels: a, large end of ulna; b, radius; c, small end of the ulna; d, carpal…

Right triangle OCA, inside of Circle O is used to show that side AC is "opposite" O and side OC is "adjacent" to O. OA is the hypotenuse. Sine is defined as the ratio of the opposite side to the hypotenuse (AC/OA). Cosine is defined as the ratio of the adjacent side to the hypotenuse (OC/OA), and Tangent is defined as the ratio of the opposite side to the adjacent side (DB/OB).

Trigonometry Triangle to Show Sine, Cosine, and Tangent

Right triangle OCA, inside of Circle O is used to show that side AC is "opposite" O and side OC is "adjacent"…

The coversedsine is equal to 1 minus the sine. In this figure, AE is parallel to OB; hence, EO = AC = sine of angle AOC, when radius OA - 1. Therefore, Coversedsine = 1 - sine = 1 - AC/OA = 1 - AC = 1 - EO = ER, when OA = 1.

Versedsine and Coversedsine in Circle

The coversedsine is equal to 1 minus the sine. In this figure, AE is parallel to OB; hence, EO = AC…

Illustration of a composite figure made up of rings, rectangles, triangles, etc..

Composite Figure

Illustration of a composite figure made up of rings, rectangles, triangles, etc..

Double Eagle ($20.00) United States coin from 1849. Obverse has a left-facing image of Liberty wearing a coronet inscribed with LIBERTY and is surrounded by 13 equally-spaced stars. Reverse shows an heraldic eagle holding arrows its left talon and an olive branch in it right. Within the ornamental scroll is inscribed E PLURIBUS UNUM and above the image is a circle of 13 stars and rays, inscription surrounding image - UNITED STATES OF AMERICA TWENTY D.

Gold Double Eagle Coin, 1849

Double Eagle ($20.00) United States coin from 1849. Obverse has a left-facing image of Liberty wearing…

Dollar ($1.00) Obverse sun over dial surrounded by the inscription - CONTINENTAL CURRENCY 1776. Reverse shows WE ARE ONE inscribed in the center surrounded by a circle around which is inscribed AMERICA CONGRESS all of which is surrounded by a series of 13 links on which are inscribed the names of the 13 states.

Pewter Dollar Coin, 1776

Dollar ($1.00) Obverse sun over dial surrounded by the inscription - CONTINENTAL CURRENCY 1776. Reverse…

Half Cent (1/2 cent) United States coin from 1809. Obverse has a left-facing profile of Liberty surrounded by 13 stars - 7-facing and is inscribed with the date - 1809. Reverse shows the value in a single olive branch bent into a circle and inscribed - UNITED STATES OF AMERICA

Copper Half Cent Coin, 1809

Half Cent (1/2 cent) United States coin from 1809. Obverse has a left-facing profile of Liberty surrounded…

Projection of a round bar.

Projection of Round Bar

Projection of a round bar.

Projection of a cylinder with a circular hole.

Projection of Cylinder With Circular Hole

Projection of a cylinder with a circular hole.

Projection of a circular ring.

Projection of Circular Ring

Projection of a circular ring.

Projections of a cylinder inclined to horizontal plane.

Projections of Cylinder

Projections of a cylinder inclined to horizontal plane.

Illustration of the method of finding the projection, in the form of an ellipse, of the top of a cylinder greatly inclined to a plane.

Projection of Cylinder

Illustration of the method of finding the projection, in the form of an ellipse, of the top of a cylinder…

Illustration showing an ellipse (and circle) as a section from a cylinder.

Section of a Cylinder Showing an Ellipse

Illustration showing an ellipse (and circle) as a section from a cylinder.

Isometric of a cube with circles inscribed on its faces.

Isometric of a Cube With Circles Inscribed

Isometric of a cube with circles inscribed on its faces.