ClipArt ETC
An illustration showing how to construct an octagon on a given line. "Prolong AB to C. With B as center and AB as radius, draw the circle AFDEC; from B, draw BI at right angles to AB; divide the angles ABC and DBC each into two equal parts; then BD is one side of the octagon. With A and E as centers, draw the arcs HKE and AKI, which determine the points H and I, and thus complete the octagon as shown in the illustration."

Construction Of An Octagon

An illustration showing how to construct an octagon on a given line. "Prolong AB to C. With B as center…

An illustration showing how to construct a regular octagon from a square by cutting off the corners of the square. "With the corners as centers, draw circle arcs through the center of the square to the side, which determine the cut-off."

Construction Of An Octagon From a Square

An illustration showing how to construct a regular octagon from a square by cutting off the corners…

An illustration showing that the area of a regular polygon is equal to the area of a triangle whose base is equal to the sum of all the sides, and the height a equal to the appotem of the polygon. "The reason of this is that the area of two or more triangles ABC and ADC having a common or equal base b and equal height h are alike."

Area Of Regular Polygon Proof

An illustration showing that the area of a regular polygon is equal to the area of a triangle whose…

An illustration showing how to construct a regular polygon on a given line without resort to its center. "Extend AB to C and, with B as center, draw the half circle ADB. Divide the half circle into as many parts as the number of sides in the polygon, and complete the construction as shown on the illustration."

Construction Of A Regular Polygon On A Line

An illustration showing how to construct a regular polygon on a given line without resort to its center.…

An illustration showing how to construct an isometric ellipse by compass and six circle arcs. "Divide OA and OB each into three equal parts; draw the quadrant AC. From C, draw the line Cc through the point 1. Through the points 2 draw de at an angle of 45° with the major axis. Then 2 is the center for the ends of the ellipse; e is the center for the arc dc; and C is the center for the arc cf."

Construction Of An Isometric Ellipse

An illustration showing how to construct an isometric ellipse by compass and six circle arcs. "Divide…

An illustration showing how to construct a hyperbola by plotting. "Having given the transverse axis BC, vertexes Aa, and foci ff'. Set off any desired number of parts on the axis below the focus, and number them 1,2,3,4,,5,etc. Take the distance a1 as radius, and, with f' as center, strike the cross 1 with f'1=a1. With the distance A1, and the focus f as center, strike the cross 1 with the radius F1=A1, and the cross 1 is a point in the hyperbola."

Construction Of A Hyperbola

An illustration showing how to construct a hyperbola by plotting. "Having given the transverse axis…

An illustration showing how to construct a hyperbola by a pencil and a string. "Having given the transverse axis BC, foci f' and f, and the vertexes A and a. Take a rule and fix it to a string at e; fix the other end of the string at the focus f. The length of the string should be such that when the rule R is in the position f'C, the loop of the string should reach to A; then move the rule on the focus f', and a pencil at P, stretching string, will trace the hyperbola."

Construction Of A Hyperbola

An illustration showing how to construct a hyperbola by a pencil and a string. "Having given the transverse…

A desk with math books and a protractor.

Math Books

A desk with math books and a protractor.

An illustration showing how to construct a parabola by plotting. "Having given the axis, vertex, and focus of the parabola. Divide the transverse axis into any desired number of parts 1, 2, 3, etc., and draw ordinates through the divisions; take the distance A1, and set it off on the 1st ordinate from the focus f to a, so that A1 = fa. Repeat the same operation with the other ordinates - that is, set off the distance A5 from f to e, so that A5 = fe; and so the parabola is constructed."

Construction Of A Parabola

An illustration showing how to construct a parabola by plotting. "Having given the axis, vertex, and…

An illustration showing how to construct a parabola using a pencil and a string. "Having given the two axes, vertex, and focus of the parabola. Take a square cde, and fix to it a string at c; fix the other end of the string at the focus f. The length of the sting should be such that when the square is in the position of the axis Af, the string should reach to the vertex A. Move the square along BB, and the pencil P will describe the parabola."

Construction Of A Parabola

An illustration showing how to construct a parabola using a pencil and a string. "Having given the two…

An illustration showing how to construct Shield's anti-friction curve. "R represents the radius of the shaft, and C1, 2, 3, et., is the center line of the shaft. From o, set off the small distance oa; and set off a1 - R. Set off the same small distance from a to b, and make b2 = R. Continue in the same way with the other points, and the anti-friction curve is thus constructed.

Construction Of Shield's Anti-friction Curve

An illustration showing how to construct Shield's anti-friction curve. "R represents the radius of the…

An illustration showing how to use isometric perspective. "This kind of perspective admits of scale measurements the same as any ordinary drawing, and gives a clear representation of the object. It is easily learned. All horizontal rectangular lines are drawn at an angle of 30°. All circles are ellipses of proportion, as shown."

Construction Using Isometric Perspective

An illustration showing how to use isometric perspective. "This kind of perspective admits of scale…

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…

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 a cycloid. "The circumference C=3.14D. Divide the rolling circle and base line C into a number of equal parts, draw through the division point the ordinates and abscissas, make aa' = 1d, bb' = 2'e, cc = 3f, then ab' and c' are points in the cycloid. In the Epicycloid and Hypocycloid the abscissas are circles and the ordinates are radii to one common center."

Construction Of A Cycloid

An illustration showing how to construct a cycloid. "The circumference C=3.14D. Divide the rolling circle…

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…

An illustration showing how to construct a spiral with compasses and four centers. "Given the pitch of the spiral, construct a square about the center, with the four sides together equal to the pitch. Prolong the sides in one direction as shown by the figure, the corners are the centers for each arc of the external angles."

Construction Of A Spiral

An illustration showing how to construct a spiral with compasses and four centers. "Given the pitch…

An illustration showing how to construct a parabola. "Given the vertex A, axis x, and a point P. Draw AB at right angle to x, and BP parallel to x, divide AB and BP into an equal number of equal parts. From the vertex A draw lines to the divisions on BP, from the divisions on AB draw the ordinates parallel to x, the corresponding intersections are points in the parabola."

Construction Of A Parabola

An illustration showing how to construct a parabola. "Given the vertex A, axis x, and a point P. Draw…

An illustration showing how to construct a parabola. "Given the axis of ordinate B, and vertex A. Take A as a center and describe a semicircle from B which gives the focus of the parabola at f. Draw any ordinate y at right angle to the abscissa Ax, take a as radius and the focus f as a center, then intersect the ordinate y, by a circle-arc in P which will be a point in the parabola. In the same manner the whole Parabola is constructed."

Construction Of A Parabola

An illustration showing how to construct a parabola. "Given the axis of ordinate B, and vertex A. Take…

An illustration showing how to construct an arithmetic spiral. "Given the pitch p and angle v, divide them into an equal number of equal parts, say 6; make 01 = 01, 02 = 02, 03 = 03, 04 = 04, 05 = 05, and 06 = the pitch p; then join the points 1, 2, 3, 4, 5 and 6, which will form the spiral required."

Construction Of An Arithmetic Spiral

An illustration showing how to construct an arithmetic spiral. "Given the pitch p and angle v, divide…

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.

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.

An illustration showing a triangle with sides b, c, and d inscribed in a circle with radius r.

Triangle Inscribed In A Circle

An illustration showing a triangle with sides b, c, and d inscribed in a circle with radius r.

An illustration showing a circle with radius r inscribed in a triangle with sides a, b, and c.

Circle Inscribed In A Triangle

An illustration showing a circle with radius r inscribed in a triangle with sides a, b, and c.

An illustration showing a quadrilateral inscribed in a circle that is tangent to a line.

Quadrilateral Inscribed In A Circle

An illustration showing a quadrilateral inscribed in a circle that is tangent to a line.

An illustration showing a triangle with interior angles A, B, C, and exterior angles D, and A' + B'.

Exterior And Interior Angles Of A Triangle

An illustration showing a triangle with interior angles A, B, C, and exterior angles D, and A' + B'.

An illustration showing a triangle with angles A, C, and D inscribed in a circle which is tangent to a line.

Triangle Inscribed In A Circle

An illustration showing a triangle with angles A, C, and D inscribed in a circle which is tangent to…

An illustration showing a circle with sectors D and E inscribed in a triangle with angles, A, B, and C.

Circle Inscribed In A Triangle

An illustration showing a circle with sectors D and E inscribed in a triangle with angles, A, B, and…

Two students drawing geometrical shapes on the chalkboard. A student is also using a scale while students sitting at desks take notes.

Geometry

Two students drawing geometrical shapes on the chalkboard. A student is also using a scale while students…

Illustration showing how to find the area of a hexagon using the triangles that make it up.

Area of Hexagon

Illustration showing how to find the area of a hexagon using the triangles that make it up.

"It is found that the cord covering the curved surface is twice as long as the one covering the flat surface. So the area of the entire curved surface of a sphere is equal to the area of the surface of 4 circles like the one measured." -Foster, 1921

Area of Sphere

"It is found that the cord covering the curved surface is twice as long as the one covering the flat…

"The bases of the pyramids are considered as forming the surface of the sphere, while the altitude of the pyramids is the radius of the sphere." -Foster, 1921

Area of Sphere

"The bases of the pyramids are considered as forming the surface of the sphere, while the altitude of…

An illustration showing how to construct a screw helix.

Construction Of A Screw Helix

An illustration showing how to construct a screw helix.

An illustration used to "square a circumference".

Square And Circle

An illustration used to "square a circumference".

An illustration used to "square a cirleplane".

Square And Circle

An illustration used to "square a cirleplane".

An illustration of an elliptical prism.

Elliptical Prism

An illustration of an elliptical prism.

An illustration of a hollow sphere.

Hollow Sphere

An illustration of a hollow sphere.

An illustration of a segment of a sphere.

Segment Of Sphere

An illustration of a segment of a sphere.

An illustration of a zone of a sphere.

Zone Of Sphere

An illustration of a zone of a sphere.

An illustration of a trapezoid with 4 sides and height labeled.

Trapezoid With Labels

An illustration of a trapezoid with 4 sides and height labeled.

Labeled shapes showing various measures of form.

Measures of Form

Labeled shapes showing various measures of form.

Showing different types of angles: right, acute, and obtuse.

Angles

Showing different types of angles: right, acute, and obtuse.

Showing different types of forms or shapes: rectangle, right triangle, acute triangle, and obtuse triangle.

Forms

Showing different types of forms or shapes: rectangle, right triangle, acute triangle, and obtuse triangle.

Showing different types of parallel prisms: rectangular, right triangular, acute triangular, and obtuse triangular.

Parallel Prisms

Showing different types of parallel prisms: rectangular, right triangular, acute triangular, and obtuse…

Showing different types of oblique prisms: rectangular, right triangular, acute triangular, and obtuse triangular.

Oblique Prisms

Showing different types of oblique prisms: rectangular, right triangular, acute triangular, and obtuse…

A rectangular prism with labeled sides: AAAA are the four vertical parallel corners. BBBB and CCCC are the horizontal parallel edges.

Rectangular Prism

A rectangular prism with labeled sides: AAAA are the four vertical parallel corners. BBBB and CCCC are…

Types of parallel prisms: right triangular prism, isosceles triangular prism, rectangular prism, cylinder.

Parallel Prisms

Types of parallel prisms: right triangular prism, isosceles triangular prism, rectangular prism, cylinder.

Types of oblique prisms: right triangular prism, isosceles triangular prism, rectangular prism, cylinder.

Oblique Prisms

Types of oblique prisms: right triangular prism, isosceles triangular prism, rectangular prism, cylinder.

Types of cylinders: vertical, horizontal, receding, and oblique.

Cylinders

Types of cylinders: vertical, horizontal, receding, and oblique.

Various geometric solids, mostly rectangular prisms with a cylinder and a hexagonal prism.

Solids

Various geometric solids, mostly rectangular prisms with a cylinder and a hexagonal prism.

Illustration showing a how in "analytic geometry it is customary to specify the position of a plane in space by giving the lengths that the plane in question cuts off from three fixed straight lines, which meet at a common point and are called 'axes.'"

Plane

Illustration showing a how in "analytic geometry it is customary to specify the position of a plane…

Illustration showing a tangent curve.

Tangent Curve

Illustration showing a tangent curve.

Illustration showing a cardioide.

Cardioide

Illustration showing a cardioide.

Illustration showing a logarithmic spiral.

Logarithmic Spiral

Illustration showing a logarithmic spiral.