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Illustration showing equilateral curves.

Equilateral Curves

Illustration showing equilateral curves.

Illustration showing a cycloid curve. "The curve generated by a point in the plane of a circle when the circle is rolled along a straight line and always in the same plane."

Cycloid

Illustration showing a cycloid curve. "The curve generated by a point in the plane of a circle when…

Illustration showing cycloid curves. "The curve generated by a point in the plane of a circle when the circle is rolled along a straight line and always in the same plane."

Cycloids

Illustration showing cycloid curves. "The curve generated by a point in the plane of a circle when the…

Illustration showing a Cassinian Oval.

Cassinian Oval

Illustration showing a Cassinian Oval.

Illustration showing a tractrix curve.

Tractrix

Illustration showing a tractrix curve.

Illustration showing a parabola with curve lines.

Parabola

Illustration showing a parabola with curve lines.

Illustration showing confocal curves.

Confocal Curves

Illustration showing confocal curves.

Illustration showing trajectory curves.

Trajectory Curves

Illustration showing trajectory curves.

Illustration showing Pascal's Volute curves.

Pascal's Volute Curves

Illustration showing Pascal's Volute curves.

Illustration showing an Archimedean Spiral.

Archimedean Spiral

Illustration showing an Archimedean Spiral.

Illustration showing a logarithmic spiral/curve.

Logarithmic Spiral

Illustration showing a logarithmic spiral/curve.

Illustration showing a lituus spiral/curve.

Lituus Spiral

Illustration showing a lituus spiral/curve.

Illustration showing conchoidal curves.

Conchoidal Curves

Illustration showing conchoidal curves.

Illustration showing conchoidal curves.

Conchoidal Curves

Illustration showing conchoidal curves.

Illustration showing a cycloid curve. "The curve generated by a point in the plane of a circle when the circle is rolled along a straight line and always in the same plane."

Cycloid

Illustration showing a cycloid curve. "The curve generated by a point in the plane of a circle when…

Trajectory is the path a moving object follows through space. A trajectory can be described mathematically either by the geometry of the path, or as the position of the object over time.

Trajectory

Trajectory is the path a moving object follows through space. A trajectory can be described mathematically…

"Nodal Cubic, with four primary lines and their satellite. In the diagram, ABC is the satellite line. From its intersections with the cubic curve tangents are drawn to the latter, AD, AE, BF, BG, CH, CI. The points of tangency lie three by three on four primary lines, FDH, DGI, EGH, FEI." -Whitney, 1911

Satellite

"Nodal Cubic, with four primary lines and their satellite. In the diagram, ABC is the satellite line.…

"In geometry: (a) A plane figure inclosed between the arc of a circle, ellipse, or other central curve and two radii to its extremities from the center. Thus, in the figure, CDB is a sector of a circle." -Whitney, 1911

Sector

"In geometry: (a) A plane figure inclosed between the arc of a circle, ellipse, or other central curve…

"Architectural Refinement from Church of St. Quentin, France ... deviations from the geometrical accuracy of purely structural lines, which have been found widely distributed in architecture before the most modern era." -Whitney, 1911

Refinement

"Architectural Refinement from Church of St. Quentin, France ... deviations from the geometrical…

"In geometry, certain transcendental curves having, in polar coordinates, equations of the form ρ = a cos b θ. A, three-leaved rose of equation ρ = a sin 3 θ." -Whitney, 1911

Rose

"In geometry, certain transcendental curves having, in polar coordinates, equations of the form ρ…

"In geometry, certain transcendental curves having, in polar coordinates, equations of the form ρ = a cos b &theta. B, three-leaved rose of equation ρ = a cos 3 &theta." -Whitney, 1911

Rose

"In geometry, certain transcendental curves having, in polar coordinates, equations of the form ρ…

"In geometry, certain transcendental curves having, in polar coordinates, equations of the form ρ = a cos b &theta. C, four-leaved rose of equation ρ = a sin 2 &theta." -Whitney, 1911

Rose

"In geometry, certain transcendental curves having, in polar coordinates, equations of the form ρ…

"In geometry, certain transcendental curves having, in polar coordinates, equations of the form ρ = a cos b &theta. D, four-leaved rose of equation ρ = a cos 2 &theta." -Whitney, 1911

Rose

"In geometry, certain transcendental curves having, in polar coordinates, equations of the form ρ…

"Grolier scroll, the flowing curved lines that surround and interlace the geometrical framework of a design for a book-cover in the style of Grolier." -Whitney, 1911

Grolier Scroll

"Grolier scroll, the flowing curved lines that surround and interlace the geometrical framework of a…

"In geometry, an angle connected with an ellipse and defined as ... angle BCL, reckoned from one determinate end, B, of the transverse axis, called the eccentric angle of the point H." -Whitney, 1911

Eccentric Angle

"In geometry, an angle connected with an ellipse and defined as ... angle BCL, reckoned from one determinate…

"In geometry, a curve generated by the motion of a point on the circumference of a circle which rolls upon the convex side of a fixed circle." -Whitney, 1911

Epicycloid

"In geometry, a curve generated by the motion of a point on the circumference of a circle which rolls…

"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…

An illustration of a triangle comprised of a tower and two lines. This illustration can be used to determine the height of the tower, the hypotenuse, and distance of the tower from the object.

Triangle with Tower

An illustration of a triangle comprised of a tower and two lines. This illustration can be used to determine…

An illustration of a triangle comprised of a church and two lines. This illustration can be used to determine the height of the church steeple, the hypotenuse, and distance of the tower from object one and two.

Triangle with Church

An illustration of a triangle comprised of a church and two lines. This illustration can be used to…

An illustration of a triangle comprised of a tree and two lines. This is an example of a problem that can be used to fine the distance of an inaccessible object without measuring elevation and whether on a horizontal plane or not.

Triangle with Tree

An illustration of a triangle comprised of a tree and two lines. This is an example of a problem that…

An illustration of a square and triangles within the square. This is an example of a problem that can be used to fine the distances of two inaccessible objects without measuring elevation and whether on a horizontal plane or not.

Square

An illustration of a square and triangles within the square. This is an example of a problem that can…

An illustration of a three triangles created with boats and a lighthouse. This is an example illustration used to fine the height of an object situated about the plane of observation, and its height above the plane.

Triangle with Lighthouse

An illustration of a three triangles created with boats and a lighthouse. This is an example illustration…

Illustration showing complex numbers with a modulus equal to unity. The lines representing these numbers terminate in points lying on the circumference of a circle whose radius is unity.

Geometric Inspection of Complex Numbers

Illustration showing complex numbers with a modulus equal to unity. The lines representing these numbers…

Illustration showing how to find the cubed roots of unity by applying DeMoivre's Theorem.

Cubed Roots of Unity

Illustration showing how to find the cubed roots of unity by applying DeMoivre's Theorem.

Illustration showing how to find the fifth roots of unity by applying DeMoivre's Theorem.

Fifth Roots of Unity

Illustration showing how to find the fifth roots of unity by applying DeMoivre's Theorem.

Illustration used, with the law of sines, to find the relation between two sides of a spherical triangle and the angles opposite.

Relationships In A Spherical Triangle

Illustration used, with the law of sines, to find the relation between two sides of a spherical triangle…

Illustration used, with the law of cosines, to find the relation between the three sides and an angle of a spherical triangle.

Relationships In A Spherical Triangle

Illustration used, with the law of cosines, to find the relation between the three sides and an angle…

Illustration used to extend the law of cosines when finding the relation between the three sides and an angle of a spherical triangle. In this case both angles b and c are greater than 90°.

Relationships In A Spherical Triangle

Illustration used to extend the law of cosines when finding the relation between the three sides and…

Illustration used to extend the law of cosines when finding the relation between the three sides and an angle of a spherical triangle. In this case angle b<90&deg; and angle c>90&deg;.

Relationships In A Spherical Triangle

Illustration used to extend the law of cosines when finding the relation between the three sides and…

Illustration of a right spherical triangle with a and b the sides, and &alpha; and &beta; the angles opposite them. Side c is the hypotenuse.

Right Spherical Triangle

Illustration of a right spherical triangle with a and b the sides, and α and β the angles…

Illustration of a right spherical triangle and the five circular parts placed in the sectors of a circle in the order in which they occur in the triangle. "The ten formulas used in the solution of spherical right triangles can be expressed by means of two rules, known as Napier's rules of circular parts."

Napier's Right Spherical Triangle

Illustration of a right spherical triangle and the five circular parts placed in the sectors of a circle…

A cube (A) has sides of 20 inches in length each, making its solid contents equal 8000 cubic inches. Being added are 3 equal portions 20x20x5, equaling 2000 cubic inches. The sum of these are 6000. You can find the second portion of the problem <a href="../62392/62392_cube_add2.htm">here</a>.

Cube with Additions 1

A cube (A) has sides of 20 inches in length each, making its solid contents equal 8000 cubic inches.…

In order to fill in the spaces from the three 2000 cubic inch additions, four new additions must be added: three 20x5x5 bars equaling 500 cubic inches and a 5x5x5 (125 cubic inches) cube for the corner. You can find the final cube <a href="../62393/62393_cube_add3.htm">here</a>.

Cube with Additions 2

In order to fill in the spaces from the three 2000 cubic inch additions, four new additions must be…

This is the final form of the original 20x20x20 inch or 8000 cubic inch cube with the addition of 7625 cubic inches making it a 25x25x25 inch cube equaling 15,625 cubic inches. You can find the original cube <a href="../62391/62391_cube_add1.htm">here</a>.

Cube with Additions 3

This is the final form of the original 20x20x20 inch or 8000 cubic inch cube with the addition of 7625…

"Folium of Descartes, with its asymptote. The equation is (4-y)(y-1)<sup>2</sup> = 3x<sup>2</sup>y ... In geometry, a plane cubic curve having a crunode, and one real inflexion, which lies at infinity.

Folium of Descartes

"Folium of Descartes, with its asymptote. The equation is (4-y)(y-1)2 = 3x2y ... In geometry, a plane…

"The part of any solid between two planes, which may be either parallel or inclined to each other: as, the frustum of a cone ... In the figure the dotted line, c, indicates the part of the cone cut off to form the frustum, f." -Whitney, 1911

Frustum of a Cone

"The part of any solid between two planes, which may be either parallel or inclined to each other: as,…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a double arch.

Double Arch

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arch.

Arch

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a triangular double arch.

Triangular Double Arch

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bear.

Bear

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a camel.

Camel

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a lit candle.

Lit Candle

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a sitting cat.

Sitting Cat

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a cat.

Cat

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a cormorant.

Cormorant

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a cup.

Cup

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a dancer.

Dancer

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a double arrow.

Double Arrow

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a diver.

Diver

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a dog.

Dog

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…