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Illustration of 36 congruent cubes stacked at various heights with outer edges forming a square. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

36 Stacked Congruent Cubes

Illustration of 36 congruent cubes stacked at various heights with outer edges forming a square. A 3-dimensional…

Illustration of 36 congruent cubes stacked to resemble a 1 by 1 by 1 cube on a 2 by 2 by 2 cube on a 3 by 3 by 3 cube. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

36 Stacked Congruent Cubes

Illustration of 36 congruent cubes stacked to resemble a 1 by 1 by 1 cube on a 2 by 2 by 2 cube on a…

Illustration of 39 congruent cubes stacked at various heights. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

39 Stacked Congruent Cubes

Illustration of 39 congruent cubes stacked at various heights. A 3-dimensional representation on a 2-dimensional…

Illustration of 4 congruent cubes stacked in ones and twos. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

4 Stacked Congruent Cubes

Illustration of 4 congruent cubes stacked in ones and twos. A 3-dimensional representation on a 2-dimensional…

Illustration of 50 congruent cubes stacked at various heights. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

50 Stacked Congruent Cubes

Illustration of 50 congruent cubes stacked at various heights. A 3-dimensional representation on a 2-dimensional…

Illustration of 56 congruent cubes stacked in twos in the shape of a square. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

56 Stacked Congruent Cubes

Illustration of 56 congruent cubes stacked in twos in the shape of a square. A 3-dimensional representation…

Illustration of 56 congruent cubes stacked in heights of 1, 4, and 5 cubes that form a zigzag pattern. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

56 Stacked Congruent Cubes

Illustration of 56 congruent cubes stacked in heights of 1, 4, and 5 cubes that form a zigzag pattern.…

Illustration of 57 congruent cubes stacked in heights of 1 and 5 cubes that form a zigzag pattern. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

57 Stacked Congruent Cubes

Illustration of 57 congruent cubes stacked in heights of 1 and 5 cubes that form a zigzag pattern. A…

Illustration of 59 congruent cubes stacked at various heights. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

59 Stacked Congruent Cubes

Illustration of 59 congruent cubes stacked at various heights. A 3-dimensional representation on a 2-dimensional…

Illustration of 64 congruent cubes stacked so they form a cube that measures 4 by 4 by 4. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

64 Stacked Congruent Cubes

Illustration of 64 congruent cubes stacked so they form a cube that measures 4 by 4 by 4. A 3-dimensional…

Illustration of 65 congruent cubes stacked at heights increasing from 1 to 5 cubes. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

65 Stacked Congruent Cubes

Illustration of 65 congruent cubes stacked at heights increasing from 1 to 5 cubes. A 3-dimensional…

Any four-point on a curve of the second order and the four-side formed by the tangents at these points stand in this relation that the diagonal points of the four-point lie in the diagonals of the four-side.

Second Order Curve Tangents

Any four-point on a curve of the second order and the four-side formed by the tangents at these points…

Illustration showing a tangent curve.

Tangent Curve

Illustration showing a tangent curve.

A large cylinder containing 2 smaller congruent cylinders. The small cylinders are externally tangent to each other and internally tangent to the larger cylinder.

2 Smaller Cylinders In A Larger Cylinder

A large cylinder containing 2 smaller congruent cylinders. The small cylinders are externally tangent…

Illustration of 3 right congruent tangent circular cylinders. The height of all the cylinders is greater than the diameter of the base.

3 Congruent Tangent Right Circular Cylinders

Illustration of 3 right congruent tangent circular cylinders. The height of all the cylinders is greater…

Illustration of 3 right congruent tangent circular cylinders.

3 Congruent Tangent Right Circular Cylinders

Illustration of 3 right congruent tangent circular cylinders.

Illustration of 3 right congruent tangent circular cylinders. The height of all the cylinders is greater than the diameter of the base.

3 Congruent Tangent Right Circular Cylinders

Illustration of 3 right congruent tangent circular cylinders. The height of all the cylinders is greater…

A large cylinder containing 3 smaller congruent cylinders. The small cylinders are externally tangent to each other and internally tangent to the larger cylinder.

3 Smaller Cylinders In A Larger Cylinder

A large cylinder containing 3 smaller congruent cylinders. The small cylinders are externally tangent…

Illustration of 4 congruent tangent right circular cylinders. The height of all the cylinders is greater than the diameter of the base.

4 Congruent Tangent Right Circular Cylinders

Illustration of 4 congruent tangent right circular cylinders. The height of all the cylinders is greater…

A large cylinder containing 4 smaller congruent cylinders. The small cylinders are externally tangent to each other and internally tangent to the larger cylinder.

4 Smaller Cylinders In A Larger Cylinder

A large cylinder containing 4 smaller congruent cylinders. The small cylinders are externally tangent…

Illustration of 4 congruent cylinders with diameters less than the height. The cylinders are externally tangent to each other.

4 Tangent Cylinders

Illustration of 4 congruent cylinders with diameters less than the height. The cylinders are externally…

A large cylinder containing 7 smaller congruent cylinders. The small cylinders are externally tangent to each other and internally tangent to the larger cylinder.

7 Smaller Cylinders In A Larger Cylinder

A large cylinder containing 7 smaller congruent cylinders. The small cylinders are externally tangent…

Illustration of 7 congruent cylinders with diameters less than the height. 6 of the cylinders are equally placed about the center cylinder. The cylinders are externally tangent to each other.

7 Tangent Cylinders

Illustration of 7 congruent cylinders with diameters less than the height. 6 of the cylinders are equally…

Illustration of 7 congruent cylinders with diameters less than the height. 6 of the cylinders are equally placed about the center cylinder. The cylinders are externally tangent to each other.

7 Tangent Cylinders

Illustration of 7 congruent cylinders with diameters less than the height. 6 of the cylinders are equally…

An illustration showing how to construct a cyma, or two circle arcs that will tangent themselves, and two parallel lines at given points A and B. "Join A and B; divide AB into four equal parts and erect perpendiculars. Draw Am at right angles from A, and Bn at right angles from B; then m and n are the centers of the circle arcs of the required cyma."

Construction Of A Cyma

An illustration showing how to construct a cyma, or two circle arcs that will tangent themselves, and…

Illustration of a 12-point star (24-sided polygon) inscribed in a regular dodecagon. This can also be described as a regular dodecagon circumscribed about a 12-point star (24-sided polygon).

12-Point Star Inscribed In A Dodecagon

Illustration of a 12-point star (24-sided polygon) inscribed in a regular dodecagon. This can also be…

Illustration of an ellipse, whose major axis is vertical, inscribed in a circle whose diameter is equal to the length of the major axis of the ellipse. The ellipse is externally tangent to the circle.

Ellipse Inscribed In A Circle

Illustration of an ellipse, whose major axis is vertical, inscribed in a circle whose diameter is equal…

Illustration of 2 concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments, are externally tangent to the circle. The illustration could be used as a 3-dimensional drawing of a sphere.

2 Ellipses Inscribed In A Circle

Illustration of 2 concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

An illustration of 2 ellipses that have the equal vertical axes, but different horizontal axes. The ellipse on the left has a larger horizontal axis than the ellipse on the right.

2 Ellipses With Equal Vertical Axes

An illustration of 2 ellipses that have the equal vertical axes, but different horizontal axes. The…

An illustration of 2 ellipses that have the equal vertical axes, but different horizontal axes. The ellipse on the left has a larger horizontal axis than the ellipse on the right. The ellipse on the left has equal horizontal and vertical axes, making it a circle.

2 Ellipses With Equal Vertical Axes

An illustration of 2 ellipses that have the equal vertical axes, but different horizontal axes. The…

An illustration of 3 concentric ellipses that are tangent at the end points of the vertical axes. The horizontal axes (the major axes of the ellipses) decreases in size in each successive ellipse. The major axis equals the minor axis in the smallest ellipse, thus forming a circle.

3 Concentric Ellipses

An illustration of 3 concentric ellipses that are tangent at the end points of the vertical axes. The…

An illustration of 3 concentric ellipses that are tangent at the end points of the vertical axes, which is drawn in the illustration. The horizontal axes decreases in size in each successive ellipse. The major axis is horizontal for the outmost ellipse and vertical for the innermost ellipse. When the major and minor axes are equal, the result is a circle (as in the second/middle ellipse).

3 Concentric Ellipses

An illustration of 3 concentric ellipses that are tangent at the end points of the vertical axes, which…

An illustration of 3 concentric ellipses that are tangent at the end points of the vertical axes. The horizontal axes decreases in size in each successive ellipse. The major axis is horizontal for the outmost ellipse and vertical for the innermost ellipse. When the major and minor axes are equal, the result is a circle (as in the second/middle ellipse).

3 Concentric Ellipses

An illustration of 3 concentric ellipses that are tangent at the end points of the vertical axes. The…

An illustration of 4 concentric ellipses that are tangent at the end points of the vertical axes. The horizontal axes (the major axes of the ellipses) decreases in size in each successive ellipse.

4 Concentric Ellipses

An illustration of 4 concentric ellipses that are tangent at the end points of the vertical axes. The…

An illustration of 4 concentric ellipses that are tangent at the end points of the vertical axes. The horizontal axes decreases in size in each successive ellipse. The major axis is horizontal for the outer two ellipses and vertical for the innermost ellipse. When the major and minor axes are equal, the result is a circle (as in the third ellipse).

4 Concentric Ellipses

An illustration of 4 concentric ellipses that are tangent at the end points of the vertical axes. The…

An illustration of 4 concentric ellipses that are tangent at the end points of the vertical axes, which is drawn in the illustration. The horizontal axes decreases in size in each successive ellipse. The major axis is horizontal for the outer two ellipses and vertical for the innermost ellipse. When the major and minor axes are equal, the result is a circle (as in the third ellipse).

4 Concentric Ellipses

An illustration of 4 concentric ellipses that are tangent at the end points of the vertical axes, which…

An illustration of 5 concentric ellipses that are tangent at the end points of the vertical axes. The horizontal axes decreases in size in each successive ellipse. The major axis is horizontal for the outer three ellipses and vertical for the innermost ellipse. When the major and minor axes are equal, the result is a circle (as in the fourth ellipse).

5 Concentric Ellipses

An illustration of 5 concentric ellipses that are tangent at the end points of the vertical axes. The…

An illustration of 5 concentric ellipses that are tangent at the end points of the vertical axes, which is drawn in the illustration. The horizontal axes decreases in size in each successive ellipse. The major axis is horizontal for the outer three ellipses and vertical for the innermost ellipse. When the major and minor axes are equal, the result is a circle (as in the fourth ellipse).

5 Concentric Ellipses

An illustration of 5 concentric ellipses that are tangent at the end points of the vertical axes, which…

An illustration of 6 concentric ellipses that are tangent at the end points of the vertical axes. The horizontal axes decreases in size in each successive ellipse. The major axis is horizontal for the outer four ellipses and vertical for the innermost ellipse. When the major and minor axes are equal, the result is a circle (as in the fifth ellipse).

6 Concentric Ellipses

An illustration of 6 concentric ellipses that are tangent at the end points of the vertical axes. The…

An illustration of 6 concentric ellipses that are tangent at the end points of the vertical axes, which is drawn in the illustration. The horizontal axes decreases in size in each successive ellipse. The major axis is horizontal for the outer four ellipses and vertical for the innermost ellipse. When the major and minor axes are equal, the result is a circle (as in the fifth ellipse).

6 Concentric Ellipses

An illustration of 6 concentric ellipses that are tangent at the end points of the vertical axes, which…

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments until the smallest one is a line, are externally tangent to the circle. The illustration could be described as a circle rotated about the poles of the vertical axis. It could also be used as a 3-dimensional drawing of a sphere.

Ellipses Inscribed In A Circle

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments until the smallest one is a line, are externally tangent to the circle. The illustration could be described as a circle rotated about the poles of the vertical axis. It could also be used as a 3-dimensional drawing of a sphere.

Ellipses Inscribed In A Circle

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

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…

"If a body be fastened to a string and whirled, so as to give it a circular motion, there will be a pull on the string that will be greater or less according as the velocity increases or decreases... If the string were cut, the pulling force that drew it away from the straight line would be removed, and the body would then fly off at a tangent; that is, it would move in a straight line tangent to the circle, as shown in Fig. 9." —Hallock 1905

Centrifugal Force

"If a body be fastened to a string and whirled, so as to give it a circular motion, there will be a…

"The galvanometer is an instrument for determining the strength of an electric current by means of the deflection of a magnetic needle around which the current flows. When a galvanoscope is provided with a scale so that the deflections of its needle may be measured, it becomes a galvanometer." — Avery, 1895

Tangent galvanometer

"The galvanometer is an instrument for determining the strength of an electric current by means of the…

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…

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…

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle circumscribed about a hexagon. In this illustration, the hexagon is not regular (the lengths of the sides are not equal).

Cyclic Hexagon

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle…

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle circumscribed about a hexagon. In this illustration, the hexagon is not regular (the lengths of the sides are not equal).

Cyclic Hexagon

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle…

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…

All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area.

Hyperbola Tangent Triangles

All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area.

Illustration that can be used to show that when given an angle, expressed as an inverse function of u, it can be used to find the value of any function of the angle in terms of u.

Angle Expressed As An Inverse Function

Illustration that can be used to show that when given an angle, expressed as an inverse function of…

The lines joining any point on a conic to the two foci are equally inclined to the tangent and normal at that point. This is an ellipse.

Conic Foci Involution

The lines joining any point on a conic to the two foci are equally inclined to the tangent and normal…

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances from two fixed points is the square of half the distance between the points. It is a particular case of the Cassinian oval and resembles a figure 8. When the line joining the two fixed points is the axis of x and the middle point of this line is the origin, the Cartesian equation is the fourth degree equation, (((x^2)+(y^2))^2)=2(a^2)((x^2)-(y^2)). The polar equation is (ℽ^2) = 2(a^2)cos(2θ). The locus of the feet of the perpendiculars from the center of an equilateral hyperbola to its tangents is a lemniscate. The name lemniscate is sometimes given to any crunodal symmetric quartic curve having no infinite branch. The name is also sometimes given to a general class of curves derived from other curves in the way that the above is derived from the equilateral hyperbola. With these more general definitions of the lemniscate the above curve is called the lemniscate of Bernoulli.

Lemniscate

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances…

The segments between the point of intersection of two tangents to a conic and their points of contact are seen from a focus under equal angles. the ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant.

Parabola Foci Properties

The segments between the point of intersection of two tangents to a conic and their points of contact…

Illustration of a tangent line drawn from an external point to a parabola.

Tangent to a Parabola

Illustration of a tangent line drawn from an external point to a parabola.

Illustration showing that the area of a parabolic segment made by a chord is two thirds the area of the triangle formed by the chord and the tangents drawn through the ends of the chord.

Tangents and Chords of a Parabola

Illustration showing that the area of a parabolic segment made by a chord is two thirds the area of…