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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…

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…

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…

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…

"Hexagonal-rhombohedral. Crystals are usually the unit rhombohedron (cleavage rhombohedron). Faces often curved, and sometimes so acurately as to form "saddle-shaped" crystals. Other forms rare." — Ford, 1912

Dolomite

"Hexagonal-rhombohedral. Crystals are usually the unit rhombohedron (cleavage rhombohedron). Faces often…

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 dolphin.

Dolphin

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 dolphin.

Dolphin

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 dolphin.

Dolphin

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 dolphin.

Dolphin

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 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 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 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 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 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 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 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 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 duck.

Duck

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 duck.

Duck

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 duck.

Duck

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 duck.

Duck

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

"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…

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 egret facing backward.

Egret Facing Backward

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 egret facing backward.

Egret Facing Backward

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 egret facing backward.

Egret Facing Backward

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 egret facing backward.

Egret Facing Backward

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 egret facing forward.

Egret Facing Forward

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 egret facing forward.

Egret Facing Forward

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 egret facing forward.

Egret Facing Forward

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 egret facing forward.

Egret Facing Forward

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

Illustration of an ellipse.

Ellipse

Illustration of an ellipse.

Illustration of an ellipse.

Ellipse

Illustration of an ellipse.

Illustration of an ellipse.

Ellipse

Illustration of an ellipse.

An illustration showing how to construct an ellipse parallel to two parallel lines A and B. "Draw a semicircle on AB, draw ordinates in the circle at right angle to AB, the corresponding and equal ordinates for the ellipse to be drawn parallel to the lines, and thus the elliptic curve is obtained as shown by the figure."

Construction Of An Ellipse Tangent To Two Parallel Lines

An illustration showing how to construct an ellipse parallel to two parallel lines A and B. "Draw a…

An illustration showing how to construct an ellipse. "With a as a center, draw two concentric circles with diameters equal to the long and short axes of the desired ellipse. Draw from o any number of radii, A, B, etc. Draw a line Bb' parallel to n and bb' parallel to m, then b is a point in the desired ellipse.

Construction Of An Ellipse

An illustration showing how to construct an ellipse. "With a as a center, draw two concentric circles…

An illustration showing how to construct an ellipse using a string. "Having given the two axes, set off from c half the great axis at a and b, which are the two focuses of the ellipse. Take an endless string as long as the three sides in the triangle abc, fix two pins or nails in the focuses, one in a and one in b, lay the string around a and b, stretch it with a pencil d, which then will describe the desired ellipse."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using a string. "Having given the two axes, set…

An illustration showing how to construct an ellipse using circle arcs. "Divide the long axis into three equal parts, draw the two circles, and where they intersect one another are the centers for the tangent arcs of the ellipses as shown by the figure."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using circle arcs. "Divide the long axis into three…

An illustration showing how to construct an ellipse using circle arcs. "Given the two axes, set off the short axis from A to b, divide b into three equal parts, set off two of these parts from o towards c and c which are the centers for the ends of the ellipse. Make equilateral triangles on cc, when ee will be the centers for the sides of the ellipse. If the long axis is more than twice the short one, this construction will not make a good ellipse."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using circle arcs. "Given the two axes, set off…

An illustration showing how to construct an ellipse. Given the two axes, set off half the long axis from c to ff, which will be the two focuses in the ellipse. Divide the long axis into any number of parts, say a to be a division point. Take Aa as radius and f as center and describe a circle arc about b, take aB as radius and f as center describe another circle arc about b, then the intersection b is a point in the ellipse, and so the whole ellipse can be constructed."

Construction Of An Ellipse

An illustration showing how to construct an ellipse. Given the two axes, set off half the long axis…

"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…

"Monoclinic. Crystals are often much elongated parallel to the ortho-axis with a prominent development of the faces of the orthodome zone, giving them a prismatic aspect. Striated parallel to the ortho-axis. Terminated usually at one end of the ortho-axis and most commonly by the two faces of a pyramid." — Ford, 1912

Epidote

"Monoclinic. Crystals are often much elongated parallel to the ortho-axis with a prominent development…

An illustration showing the construction used to erect an equal angle. "With D as a center, draw the dotted arc CE: and with the same radius and B as a center, draw the arc GF; then make GF equal to CE; then join BF, which will form the required angle, FBG=CDE."

Construction Of An Equal Angle

An illustration showing the construction used to erect an equal angle. "With D as a center, draw the…

Illustration used to show how to divide a given straight line into any number of equal parts.

Line Divided Into Equal Parts

Illustration used to show how to divide a given straight line into any number of equal parts.

Tangent lines are drawn from a given external point to a circle. The equation is found by joining their points of contact

Chord Equation

Tangent lines are drawn from a given external point to a circle. The equation is found by joining their…

Finding the equation of a circle with the origin as its center.

Circle Equation

Finding the equation of a circle with the origin as its center.

Finding the equation of a line through two given points.

Line Equation

Finding the equation of a line through two given points.

Finding the equation of the normal at any point on a circle.

Normal Equation

Finding the equation of the normal at any point on a circle.

Finding the polar equation of a straight line.

Polar Equation

Finding the polar equation of a straight line.

Other forms of line equations.

Line Equations

Other forms of line equations.

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…

Euclid, a great mathematician, wrote the first systematic textbook on geometry.

Euclid

Euclid, a great mathematician, wrote the first systematic textbook on geometry.

An illustration showing how to construct an evolute of a circle. "Given the pitch p, the angle v, and radius r. Divide the angle v into a number of equal parts, draw the radii and tangents for each part, divide the pitch p into an equal number of equal parts, then the first tangent will be one part, second two parts, third three parts, etc., and so the Evolute is traced."

Construction Of An Evolute Of A Circle

An illustration showing how to construct an evolute of a circle. "Given the pitch p, the angle v, and…

Polar property example.

Example

Polar property example.

Example of an equation with a given external point.

Equation Example

Example of an equation with a given external point.

Illustration used to show how "to construct a common external tangent to two given circles."

Construction of an External Tangent to 2 Circles

Illustration used to show how "to construct a common external tangent to two given circles."

Illustration used to show how to construct two common external tangents to two given circles.

Construction of an 2 External Tangents to 2 Circles

Illustration used to show how to construct two common external tangents to two given circles.

"This is a form consisting of six rectangular vertical faces each of which intersects two of the horizontal axes equally and the intermediate horizontal axis at one-half this distance." — Ford, 1912

Prism of the first order

"This is a form consisting of six rectangular vertical faces each of which intersects two of the horizontal…

"This form consists of twelve isoceles triangular faces, each of which intersects two of the horizontal crystallographic axes equally, is parallel to the third horizontal axis and intersects the vertical axis." — Ford, 1912

Pyramid of the first order

"This form consists of twelve isoceles triangular faces, each of which intersects two of the horizontal…

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 fisherman.

Fisherman

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