Illustration of 3 equal dihedral angles. "Two dihedral angles have the same ratio as their plane angles."

Equal Dihedral Angles

Illustration of 3 equal dihedral angles. "Two dihedral angles have the same ratio as their plane angles."

"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,…

"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,…

Illustration of equal circles to show that two central angles have the same ratio as their intercepted arcs.

Equal Circles With Intercepted Arcs

Illustration of equal circles to show that two central angles have the same ratio as their intercepted…

Illustration of 2 right circular cones that are similar.

2 Similar Right Circular Cones

Illustration of 2 right circular cones that are similar.

Illustration of 2 right circular cylinders in which one cylinder is twice the height of the other.

2 Right Circular Cylinders

Illustration of 2 right circular cylinders in which one cylinder is twice the height of the other.

Illustration of 2 right circular cylinders in which one cylinder is twice the height and twice the diameter of the other.

2 Right Circular Cylinders

Illustration of 2 right circular cylinders in which one cylinder is twice the height and twice the diameter…

Illustration of 2 similar cylinders. The height and diameter of the smaller cylinder is half that of the larger one.

2 Similar Cylinders

Illustration of 2 similar cylinders. The height and diameter of the smaller cylinder is half that of…

Illustration of 2 soup cans that are similar cylinders. The diameter and height of the smaller can is one half that of the larger.

2 Soup Can Cylinders

Illustration of 2 soup cans that are similar cylinders. The diameter and height of the smaller can is…

Illustration of 3 similar cylinders. The height and diameter in each successively smaller cylinder is 1/2 that of the previous one.

3 Similar Cylinders

Illustration of 3 similar cylinders. The height and diameter in each successively smaller cylinder is…

Illustration of 3 similar cylinders. The height and diameter in each successively smaller cylinder is one half that of the previous one.

3 Similar Cylinders

Illustration of 3 similar cylinders. The height and diameter in each successively smaller cylinder is…

2 circles that are similar figures

Similar Figures

2 circles that are similar figures

2 squares that are similar figures

Similar Figures

2 squares that are similar figures

2 rectangles that are similar figures

Similar Figures

2 rectangles that are similar figures

2 pentagons that are similar figures

Similar Figures

2 pentagons that are similar figures

An illustration showing a model of a circle with intersecting chords that illustrates the following relationship: a:c = b:d, ad = bc. Product of the means equals the product of the extremes.

Model Of Geometric Proportions In A Circle

An illustration showing a model of a circle with intersecting chords that illustrates the following…

An illustration showing a model of a triangle that illustrates the following relationship: a:c = d:(b - d), d = (a × b) ÷ (c + a), v = v.

Model Of Geometric Proportions In A Triangle

An illustration showing a model of a triangle that illustrates the following relationship: a:c = d:(b…

An illustration showing a quadrilateral model that illustrates the following relationships: a:b = c:d, ad = bc, A = B. Product of the means equals the product of the extremes.

Model Of Geometric Proportions

An illustration showing a quadrilateral model that illustrates the following relationships: a:b = c:d,…

An illustration showing a model that illustrates the following relationships: a:b = c:d, ad = bc. Product of the means equals the product of the extremes.

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationships: a:b = c:d, ad = bc. Product…

An illustration showing a model that illustrates the following relationships: a:b = c:b, ab = c², c = √ab.

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationships: a:b = c:b, ab = c²,…

An illustration showing a model that illustrates the following relationship: A:B = a:b.

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationship: A:B = a:b.

An illustration showing a model that illustrates the following relationships: a:x = x:a - x, x = √(a&sup2 + (a/2)&sup2 - a/2).

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationships: a:x = x:a - x, x = √(a²…

An illustration showing a model of 2 circles with tangent lines, diameters, and radii that illustrates the following geometric relationship: "x = aR/(R - r), a = √(t&sup2 + (R - r)&sup2), t = √(a&sup2 - (R - r)&sup2, sin.v = t/a."

Model Of Geometric Relationships In 2 Circles

An illustration showing a model of 2 circles with tangent lines, diameters, and radii that illustrates…

An illustration showing a model of 2 circles with tangent lines, diameters, and radii that illustrates the following geometric relationship: " t = √(a&sup2 - (R + r)&sup2, a = √(t&sup2 - (R + r)&sup2 "

Model Of Geometric Relationships In 2 Circles

An illustration showing a model of 2 circles with tangent lines, diameters, and radii that illustrates…

An illustration showing a model of a circle with an exterior angle formed between a tangent and a secant that illustrates the following geometric relationship: a:t = t:b, t&sup2 = ab

Model Of Geometric Relationships In A Circle

An illustration showing a model of a circle with an exterior angle formed between a tangent and a secant…

An illustration showing a model of a circle with angles formed between tangents and secants that illustrates the following geometric relationship: t&sup2 = (a + b)(a - b).

Model Of Geometric Relationships In A Circle

An illustration showing a model of a circle with angles formed between tangents and secants that illustrates…

"The proportions of the human figure. As handed down to us by Vitruvius and described by Joseph Bonomi." —D'Anvers, 1895

Proportions of human figure

"The proportions of the human figure. As handed down to us by Vitruvius and described by Joseph Bonomi."…

Illustration of an extension ladder. If ladder is leaned against a building, it will form a right triangle with the ground.

Extension Ladder

Illustration of an extension ladder. If ladder is leaned against a building, it will form a right triangle…

Illustration of a ladder that is not perpendicular to the ground. If it is set on the ground and leaned toward a building, it will form the hypotenuse of a right triangle.

Leaning Ladder

Illustration of a ladder that is not perpendicular to the ground. If it is set on the ground and leaned…

Illustration of a ladder that is not perpendicular to the ground. If it is set on the ground and leaned toward a building, it will form the hypotenuse of a right triangle.

Leaning Ladder

Illustration of a ladder that is not perpendicular to the ground. If it is set on the ground and leaned…

Illustration of a ladder that is not perpendicular to the ground. If it is set on the ground and leaned toward a building, it will form the hypotenuse of a right triangle.

Leaning Ladder

Illustration of a ladder that is not perpendicular to the ground. If it is set on the ground and leaned…

Illustration of a ladder that is not perpendicular to the ground. If it is set on the ground and leaned toward a building, it will form the hypotenuse of a right triangle.

Leaning Ladder

Illustration of a ladder that is not perpendicular to the ground. If it is set on the ground and leaned…

Illustration of a ladder leaning against the side of a building (wall) to form a right triangle .

Ladder Leaning Against a Building

Illustration of a ladder leaning against the side of a building (wall) to form a right triangle .

Illustration of a ladder leaning against a palm tree, that is perpendicular to the ground, to form a right triangle .

Ladder Leaning Against a Tree

Illustration of a ladder leaning against a palm tree, that is perpendicular to the ground, to form a…

Illustration of a ladder leaning against a palm tree, that is perpendicular to the ground, to form a right triangle .

Ladder Leaning Against a Tree

Illustration of a ladder leaning against a palm tree, that is perpendicular to the ground, to form a…

Illustration of a ladder leaning against a palm tree, that is perpendicular to the ground, to form a right triangle .

Ladder Leaning Against a Tree

Illustration of a ladder leaning against a palm tree, that is perpendicular to the ground, to form a…

Illustration of a ladder leaning against a palm tree, that is perpendicular to the ground, to form a right triangle .

Ladder Leaning Against a Tree

Illustration of a ladder leaning against a palm tree, that is perpendicular to the ground, to form a…

Illustration of 2 ladders leaning against opposite sides of a palm tree to form similar right triangles. The angles of elevation from the ground to where the ladders meet the tree are congruent. Illustration can be used for problems involving proportions.

2 Ladders Leaning Against a Tree

Illustration of 2 ladders leaning against opposite sides of a palm tree to form similar right triangles.…

Illustration of 3 ladders leaning against the side of a building (wall) to form right triangles. The distance from the base of the ladders to the wall is the same for all three ladders.

3 Ladders Leaning Against a Wall

Illustration of 3 ladders leaning against the side of a building (wall) to form right triangles. The…

An illustration of a lever with weights m and n with distances a and b from fulcrum. Illustration could be used with proportions.

Lever Balanced on fulcrum With Weights

An illustration of a lever with weights m and n with distances a and b from fulcrum. Illustration could…

Illustration that shows if two parallel lines are cut by three or more transversals that pass through the same point, the corresponding segments are proportional.

Parallel Lines Cut by Transversals

Illustration that shows if two parallel lines are cut by three or more transversals that pass through…

Illustration that shows if two parallel lines are cut by three or more transversals that pass through the same point, the corresponding segments are proportional.

Parallel Lines Cut by Transversals

Illustration that shows if two parallel lines are cut by three or more transversals that pass through…

Illustration of two straight lines that can be used to find ratios.

Straight Lines to Find Ratios

Illustration of two straight lines that can be used to find ratios.

Diagram used to prove the theorem: "The rectangular parallelopipeds which have two dimensions in common are to each other as their third dimension."

Relationship Between 2 Parallelopipeds With Equal Altitudes

Diagram used to prove the theorem: "The rectangular parallelopipeds which have two dimensions in common…

Illustration of 2 similar pentagons.

Similar Pentagons

Illustration of 2 similar pentagons.

Illustration of 2 similar pentagons.

Similar Pentagons

Illustration of 2 similar pentagons.

Diagram used to prove the theorem: "If a pyramid is cut by a plane parallel to the base, the edges are divided proportionally, and the section is a polygon similar to the base."

Pyramid Cut By Plane

Diagram used to prove the theorem: "If a pyramid is cut by a plane parallel to the base, the edges are…

"If two straight lines are cut by three parallel planes, the corresponding segments are proportional."

3 Parallel Planes

"If two straight lines are cut by three parallel planes, the corresponding segments are proportional."

Illustration that shows similar polygons (pentagons) that can be used to show proportionality.

Similar Polygons (Pentagons) That Can Be Used To Show Proportionality

Illustration that shows similar polygons (pentagons) that can be used to show proportionality.

Illustration that shows similar polygons (pentagons) that can be used to show proportionality.

Similar Polygons (Pentagons) That Can Be Used To Show Proportionality

Illustration that shows similar polygons (pentagons) that can be used to show proportionality.

Illustration of 2 right octagonal prisms with congruent bases, but different heights. The height of the smaller prism is one half that of the larger.

2 Octagonal Prisms

Illustration of 2 right octagonal prisms with congruent bases, but different heights. The height of…

Illustration of 2 right rectangular prisms. The bases are congruent, but the height of the smaller prism is one half that of the larger. Hidden edges are shown.

2 Rectangular Prisms

Illustration of 2 right rectangular prisms. The bases are congruent, but the height of the smaller prism…

Illustration of 2 right rectangular prisms. The bases are congruent, but the height of the smaller prism is one half that of the larger.

2 Rectangular Prisms

Illustration of 2 right rectangular prisms. The bases are congruent, but the height of the smaller prism…

Illustration of 2 Similar right octagonal prisms. The height and length of the edges of the smaller prism are one half that of the larger.

2 Similar Octagonal Prisms

Illustration of 2 Similar right octagonal prisms. The height and length of the edges of the smaller…

Illustration of 2 similar right decagonal prisms. Both have regular decagons for bases and rectangular faces. The height of the prism and length of the side of the decagon on the smaller decagonal prism are one half that of the larger.

Similar Decagonal Prisms

Illustration of 2 similar right decagonal prisms. Both have regular decagons for bases and rectangular…

Illustration of 2 similar right heptagonal/septagonal prisms. Both have regular heptagons/septagons for bases and rectangular faces. The height of the prism and length of the side of the heptagon on the smaller heptagonal prism are one half that of the larger.

Similar Heptagonal/Septagonal Prisms

Illustration of 2 similar right heptagonal/septagonal prisms. Both have regular heptagons/septagons…

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of the hexagon on the smaller hexagonal prism are one half that of the larger.

Similar Hexagonal Prisms

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of…

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of the hexagon on the smaller hexagonal prism are one half that of the larger.

Similar Hexagonal Prisms

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of…

An illustration showing the construction used to divide the line AB in the same proportion of parts as AC. "Join C and B, and through the given divisions 1, 2, and 3 draw lines parallel with CB, which solves the problem."

Divide A Line Proportionately

An illustration showing the construction used to divide the line AB in the same proportion of parts…