Illustration showing a cardioide.

Cardioide

Illustration showing a cardioide.

Aronhold stated if any three bodies have plane motion their three virtual centers are three points on one straight line

Centrodes

Aronhold stated if any three bodies have plane motion their three virtual centers are three points on…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. The circle is divided into four quadrants by the x- and y- axes. The circle can be labeled and used to find the six trigonometric values (sin, cos, tan, cot, sec, csc, cot) at each of the quadrantal angles.

Unit Circle

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. The…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles are given in radian measure in terms of pi.

Unit Circle Labeled At Quadrantal Angles

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles and angles that have reference angles of 30°, 45°, and 60° are given in radian measure in terms of pi. At each quadrantal angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled At Special Angles

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles and angles that have reference angles of 30°, 45°, and 60° are given in radian measure in terms of pi.

Unit Circle Labeled At Special Angles

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At 30° increments, the angles are given in both radian and degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 30° Increments With Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in both radians and degrees in 45° increments. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 45 ° Increments

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At 45° increments, the angles are given in both radian and degree measure. At each quadrantal angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 45° Increments With Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At 45° increments, the angles are given in both radian and degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 45° Increments With Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. All quadrantal angles are given in radian measure in terms of pi.  At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. All…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles are given in degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles are given in both radian and degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. At each quadrantal angle, the coordinates are given, but not the angle measure. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. At each…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At each quadrantal angle, the coordinates are given, but not the angle measure. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in radians. All quadrantal angles and angles that have reference angles of 30°, 45°, and 60° are given in radian measure in terms of pi. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Special Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in both radians and degrees at all quadrantal angles and angles that have reference angles of 30°, 45°, and 60°. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Special Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in both radians and degrees at all quadrantal angles and angles that have reference angles of 30°, 45°, and 60°. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Special Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. All quadrantal angles and angles that have reference angles of 30°, 45°, and 60° are marked from the origin, but no values are given.

Unit Circle Marked At Special Angles

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. All…

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…

Instantaneous axis of two cones, each with angular velocity

Conic Motion

Instantaneous axis of two cones, each with angular velocity

Illustration showing a tangent curve.

Tangent Curve

Illustration showing a tangent curve.

Illustration showing conchoidal curves.

Conchoidal Curves

Illustration showing conchoidal curves.

Illustration showing conchoidal curves.

Conchoidal Curves

Illustration showing conchoidal curves.

Illustration showing confocal curves.

Confocal Curves

Illustration showing confocal curves.

Illustration showing equilateral curves.

Equilateral Curves

Illustration showing equilateral curves.

Illustration showing Pascal's Volute curves.

Pascal's Volute Curves

Illustration showing Pascal's Volute curves.

Illustration showing trajectory curves.

Trajectory Curves

Illustration showing trajectory 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 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…

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…

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…

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…

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 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…

Illustration showing a Cassinian Oval.

Cassinian Oval

Illustration showing a Cassinian Oval.

Illustration showing a parabola with curve lines.

Parabola

Illustration showing a parabola with curve lines.

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…

An illustration showing how to construct a screw helix.

Construction Of A Screw Helix

An illustration showing how to construct a screw helix.

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…

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 showing an Archimedean Spiral.

Archimedean Spiral

Illustration showing an Archimedean Spiral.

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 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…

Illustration showing a lituus spiral/curve.

Lituus Spiral

Illustration showing a lituus spiral/curve.

Illustration showing a logarithmic spiral.

Logarithmic Spiral

Illustration showing a logarithmic spiral.

Illustration showing a logarithmic spiral/curve.

Logarithmic Spiral

Illustration showing a logarithmic spiral/curve.

Illustration showing a tractrix curve.

Tractrix

Illustration showing a tractrix curve.

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.