"Break lathes can take in work of considerable length as well as of large diameter,—the treble-geared headstock and all other parts being in their case made of sufficient strength to bear the heavy strains which result from the increased size and weight of the work, a quality in which gap lathes are not unfrequntly deficient." —The Encyclopedia Britannica, 1903

Break Lathe

"Break lathes can take in work of considerable length as well as of large diameter,—the treble-geared…

"A kind of compass for measuring the diameter of round bodies." — Williams, 1889

Calipers

"A kind of compass for measuring the diameter of round bodies." — Williams, 1889

Illustration of circle with diameter.

Circle with Diameter

Illustration of circle with diameter.

If every diameter is perpendicular to its conjugate, the conic is a circle.

Circle Diameters

If every diameter is perpendicular to its conjugate, the conic is a circle.

A circle with labels for radius, diameter, and circumference. The visual will help to remember what each term means.

Circle Parts

A circle with labels for radius, diameter, and circumference. The visual will help to remember what…

Illustration of a circle with an inscribed angle that can be used to prove that "An inscribed angle is measured by one half of its intercepted arc." In this case, one side of angle ABC passes through the center of the circle.

Inscribed Angle in a Circle Proof

Illustration of a circle with an inscribed angle that can be used to prove that "An inscribed angle…

Equal circles inside and tangent to the outside circle, also tangent to each other

Circle to Circle Tangents

Equal circles inside and tangent to the outside circle, also tangent to each other

Illustration of circle with 10 inch diameter and square.

Circle With 10 inch Diameter and Steel Square

Illustration of circle with 10 inch diameter and square.

Illustration of circle with 6 inch diameter.

Circle With 6 inch Diameter

Illustration of circle with 6 inch diameter.

Illustration of circle with 8 inch diameter.

Circle With 8 inch Diameter

Illustration of circle with 8 inch diameter.

An illustration used to show how the area of a circle is calculated. Area is equal to the product of pi and the radius squared.

Area Of A Circle

An illustration used to show how the area of a circle is calculated. Area is equal to the product of…

Illustration of a circle with diameter AB. Radii are RO, SO, AO, and BO. ED is a chord.

Chords, Diameters, and Radii of a Circle

Illustration of a circle with diameter AB. Radii are RO, SO, AO, and BO. ED is a chord.

Illustration of a circle with diameter BC, chord DF, secant MN, and tangent HK.

Chords, Secants, Diameters, and Tangents of a Circle

Illustration of a circle with diameter BC, chord DF, secant MN, and tangent HK.

Illustration showing a circle with a diameter, radius, and segment drawn.

Circle With Diameter, Radius, Segment

Illustration showing a circle with a diameter, radius, and segment drawn.

Illustration showing a circle with a diameter, radius, lines, triangle, and segment drawn.

Circle With Diameter, Radius, Segment, Line

Illustration showing a circle with a diameter, radius, lines, triangle, and segment drawn.

Illustration of a circle with two intersecting diameters.

Diameters of a Circle

Illustration of a circle with two intersecting diameters.

Circle with diameter, radius, arc, chord, and arc.

Circle With Parts

Circle with diameter, radius, arc, chord, and arc.

Illustration of circle with arc, chord, diameter  and radius.

Parts of Circle

Illustration of circle with arc, chord, diameter and radius.

Illustration of circle with perpendicular diameters.

Circle With Perpendicular Diameters

Illustration of circle with perpendicular diameters.

Illustration of circle with perpendicular diameters.

Circle With Perpendicular Diameters

Illustration of circle with perpendicular diameters.

"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,…

Illustration of circle with radius of 4000, and triangle with 45 degree angle enclosed.

Circles With Radius 4000 and Enclosed Triangle

Illustration of circle with radius of 4000, and triangle with 45 degree angle enclosed.

An illustration showing a circle with radius r, diameter d, and chord c.

Radius, Diameter, and Chord In A Circle

An illustration showing a circle with radius r, diameter d, and chord c.

Illustration of a regular hexagon inscribed in a circle. This can also be described as a circle circumscribed about a regular hexagon. All diagonals of the hexagon are also diameters of the circle. The diagonals intersect at the center of both the hexagon and the circle.

Regular Hexagon Inscribed In A Circle

Illustration of a regular hexagon inscribed in a circle. This can also be described as a circle circumscribed…

Illustration of circle with diameter and segment used to find area.

Circle With Diameter and Segment labeled

Illustration of circle with diameter and segment used to find area.

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 inscribed in a circle. This can also be described as a circle circumscribed about a square. The diagonal of the square is also the diameter of the circle.

Square Inscribed In A Circle

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

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 showing a circle with various chords drawn.

Circle With Various Chords

Illustration showing a circle with various chords drawn.

Illustration showing a circle with various chords drawn.

Circle With Various Chords

Illustration showing a circle with various chords drawn.

Illustration showing a circle with various chords and tangents drawn.

Circle With Various Chords And Tangents

Illustration showing a circle with various chords and tangents drawn.

Illustration showing a circle with various chords and tangents drawn.

Circle With Various Chords And Tangents

Illustration showing a circle with various chords and tangents drawn.

Illustration of three circles enclosed in an equilateral triangle.

Circles Enclosed in Equilateral Triangle

Illustration of three circles enclosed in an equilateral triangle.

Illustration of large circle and square with 4 smaller circles inside.

Circles Inside of a Square

Illustration of large circle and square with 4 smaller circles inside.

Illustration of box filled with pipes.

Circular Pipes in Box

Illustration of box filled with pipes.

"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 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.

"Epicycloidal wheel, a wheel or ring fixed to a framework, toothed on its inner side, and having in gear with it another toothed wheel, of half the diameter of the first, fitted so as to revolve about the center of the latter. It is used for converting circular into alternate motion, or alternate into circular." -Whitney, 1911

Epicycloidal Wheel

"Epicycloidal wheel, a wheel or ring fixed to a framework, toothed on its inner side, and having in…

Three suspended concentric circles free to move independently of each other at right angles.

Gyroscope

Three suspended concentric circles free to move independently of each other at right angles.

Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.

Hyperbola Conjugate Diameters

Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.

"The principle of increased velocity as bodies descend from a height, is curiously illustrated by pouring molasses or thick syrup from an elevation to the ground. The bulky stream, of perhaps two inches in diameter where it leaves the vessel, as it descends, is reduced to the size of a straw, or knitting-needle; but what it wants in bulk is made up in velocity, for the small stream at the ground will fill a vessel just as soon as the large one at the outlet." —Comstock, 1850

Increased Velocity

"The principle of increased velocity as bodies descend from a height, is curiously illustrated by pouring…

The magic circle of circles, first developed by Benjamin Franklin, consists of eight annular rings and a central circle, each ring being divided into eight cells by radii drawn from the centre; there are therefore 65 cells. The number 12 is placed in he center and the consecutive numbers 13 to 75 are placed in the other cells. The properties are: 1) sum of eight numbers in any ring with 2 equals 360, 2) sum of eight numbers in any set of radial rings with 12 is 360, 3) sum of numbers in any four adjoining cells is 180.

Magic Circle of Circles

The magic circle of circles, first developed by Benjamin Franklin, consists of eight annular rings and…

Illustration of a circle with center O and diameters AB and CD perpendicular to each other.

Circle With 2 Perpendicular Diameters

Illustration of a circle with center O and diameters AB and CD perpendicular to each other.

An illustration of a ring (circle within circle), with diameters of 3a-7 and 6a+9. Illustration could be used to calculate area.

Ring Made of Circles With Diameters 3a-7 and 6a+9

An illustration of a ring (circle within circle), with diameters of 3a-7 and 6a+9. Illustration could…

Illustration of ring (small circle in larger concentric circle).

Ring Made of Concentric Circles

Illustration of ring (small circle in larger concentric circle).

Illustration of ring (small circle in larger concentric circle) with piece cut out.

Ring With Piece Cut Out

Illustration of ring (small circle in larger concentric circle) with piece cut out.

Illustration of ring (small circle in larger concentric circle) sprung into place.

Ring Sprung Into Place

Illustration of ring (small circle in larger concentric circle) sprung into place.

An illustration showing a circle sector with radius r, center/central angle v, and length of circle arc b.

Circle Sector

An illustration showing a circle sector with radius r, center/central angle v, and length of circle…

An illustration showing a circle sector with center/central angle v and polygon angle w.

Circle Sector

An illustration showing a circle sector with center/central angle v and polygon angle w.

An illustration showing a circle sector with height of segment h and radius r.

Circle Sector

An illustration showing a circle sector with height of segment h and radius r.

Semicircle with line drawn perpendicular to the diameter.

Semicircle

Semicircle with line drawn perpendicular to the diameter.

Illustration of a semicircle.

Semicircle

Illustration of a semicircle.

Illustration of a semicircle with chords and radii.

Semicircle With Chords and Radii

Illustration of a semicircle with chords and radii.

An 8-inch sphere cut by parallel planes, one 2 inches from center and the other 6 inches from center. This illustration can be used to find the area of the zone and the volume of the segments.

Sphere With 8-inch Diameter Cut by Planes

An 8-inch sphere cut by parallel planes, one 2 inches from center and the other 6 inches from center.…

An illustration of a sphere cut into polygons as bases with their vertices at the center of sphere.

Sphere Cut Into Pyramids.

An illustration of a sphere cut into polygons as bases with their vertices at the center of sphere.

An illustration of a sphere inside of a cylinder with radius/diameter labeled. Illustration for showing how volume is found.

Sphere Inside of Cylinder.

An illustration of a sphere inside of a cylinder with radius/diameter labeled. Illustration for showing…

An illustration of a sphere showing diameter, arcs, and circles. A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within, called the center.

Sphere with arcs and circles labeled.

An illustration of a sphere showing diameter, arcs, and circles. A sphere is a solid bounded by a curved…

An illustration of a sphere showing diameter, arcs, and circles. A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within, called the center. This shows the zones of a sphere.

Sphere With Arcs, Circles, and Zones Labeled.

An illustration of a sphere showing diameter, arcs, and circles. A sphere is a solid bounded by a curved…