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From Coleridge's The Rime of the Ancient Mariner. An old mariner tells his tales to a young man who is on his way to a wedding.

Ancient Mariner

From Coleridge's The Rime of the Ancient Mariner. An old mariner tells his tales to a young man who…

Bone points and bone needles. Crafted during the Reindeer age.

Reindeer age articles, Bone points and needles

Bone points and bone needles. Crafted during the Reindeer age.

A cathedral with pointed towers.

Cathedral

A cathedral with pointed towers.

The board of trade sign for 1/8 cent.

1/8 Cent

The board of trade sign for 1/8 cent.

"The part of the body in which the centre of gravity is situated, may be found, in some cases, by balancing it on a point. Thus the centre of gravity of the poker represented [here] lies directly over the point on which it is balanced." —Quackenbos 1859

Center of Gravity

"The part of the body in which the centre of gravity is situated, may be found, in some cases, by balancing…

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height of the point P above the surface; it is required to find the distance from the point P to the horizon at A.

Circle With Center o and Radius r with point P

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height…

Illustration of a circle which illustrates that the tangents to a circle drawn from an external point are equal, and make equal angles with the line joining the point to the center.

Circle With Two Tangents Drawn From an External Point

Illustration of a circle which illustrates that the tangents to a circle drawn from an external point…

Illustration used to construct a circle when given two points that it passes through and a radius.

Construction of a Circle When Given Two Points and a Radius

Illustration used to construct a circle when given two points that it passes through and a radius.

Illustration of the construction of a perpendicular to a line when given a point O on the straight line.

Construction of Perpendicular From a Given Point on a Straight Line

Illustration of the construction of a perpendicular to a line when given a point O on the straight line.

Illustration of the construction of a perpendicular to a line when given a point B on the straight line.

Construction of Perpendicular From a Given Point on a Straight Line

Illustration of the construction of a perpendicular to a line when given a point B on the straight line.

Coordinate axis with angle XOP equal to theta, Θ, and angle XOQ=180 - Θ. From any point in the terminal side of XOP, as B, a perpendicular can be drawn, AB, to the x-axis; and from D, any point in the terminal side o f XOQ, perpendicular CD can be drawn to the x-axis. The right triangles OAB and OCD are similar. Also, OA, AB, OB, CD, and OD are positive, while OC is negative.

Coordinate Axis With Angles, Lines, and Perpendiculars Drawn

Coordinate axis with angle XOP equal to theta, Θ, and angle XOQ=180 - Θ. From any point in the terminal…

Angle XOP=Θ and angle XOQ=- Θ. From a point in the terminal side of each a perpendicular line is drawn to the x-axis. The right triangles OAB and OAC thus formed are similar, and have all their sides positive except AC, which is negative.

Coordinate Axis With Perpendiculars Drawn To Form Similar Right Triangles From Positive and Negative Theta, Θ

Angle XOP=Θ and angle XOQ=- Θ. From a point in the terminal side of each a perpendicular line is drawn…

Angle XOP=Θ and angle XOQ=90+Θ. From a point in the terminal side of each a perpendicular line is drawn to the x-axis. The right triangles AOB and OCD thus formed are similar, and have all their sides positive except OC

Coordinate Axis With Perpendiculars Drawn To Form Similar Right Triangles

Angle XOP=Θ and angle XOQ=90+Θ. From a point in the terminal side of each a perpendicular line is…

Illustration of blueprint used by highway engineers to widen the pavement on the inside of the curve of a road.

Curve in Pavement of Road

Illustration of blueprint used by highway engineers to widen the pavement on the inside of the curve…

Illustration 1 of the Dedekind axiom.

Dedekind Property 1

Illustration 1 of the Dedekind axiom.

Illustration 2 of the Dedekind axiom.

Dedekind Property 2

Illustration 2 of the Dedekind axiom.

Illustration 3 of the Dedekind axiom.

Dedekind Property 3

Illustration 3 of the Dedekind axiom.

Illustration 4 of the Dedekind axiom.

Dedekind Property 4

Illustration 4 of the Dedekind axiom.

To determine the distance between two points A, B given by their projections.

Point Distance

To determine the distance between two points A, B given by their projections.

A four-pointed doodad.

Doodad

A four-pointed doodad.

A four-pointed doodad.

Doodad

A four-pointed doodad.

A four pointed doodad.

Doodad

A four pointed doodad.

The Double Arrow sign may be used to advise road users that traffic is permitted to pass on either side of an island, obstruction, or gore in the roadway. Traffic separated by this sign may either rejoin or change directions.

Double Arrow, Black and White

The Double Arrow sign may be used to advise road users that traffic is permitted to pass on either side…

The Double Arrow sign may be used to advise road users that traffic is permitted to pass on either side of an island, obstruction, or gore in the roadway. Traffic separated by this sign may either rejoin or change directions.

Double Arrow, Color

The Double Arrow sign may be used to advise road users that traffic is permitted to pass on either side…

The Double Arrow sign may be used to advise road users that traffic is permitted to pass on either side of an island, obstruction, or gore in the roadway. Traffic separated by this sign may either rejoin or change directions.

Double Arrow, Outline

The Double Arrow sign may be used to advise road users that traffic is permitted to pass on either side…

The Double Arrow sign may be used to advise road users that traffic is permitted to pass on either side of an island, obstruction, or gore in the roadway. Traffic separated by this sign may either rejoin or change directions.

Double Arrow, Silhouette

The Double Arrow sign may be used to advise road users that traffic is permitted to pass on either side…

People driving in a car arrive at their destination and read the sign.

Driving a Car

People driving in a car arrive at their destination and read the sign.

Illustration of half of an ellipse. The square of the ordinate of a point in an ellipse is to the product of the segments of the major axis made by the ordinate as the square of b to the square of a.

Ordinate and Major Axis of Ellipse

Illustration of half of an ellipse. The square of the ordinate of a point in an ellipse is to the product…

A flashcard featuring an illustration of a Point

Flashcard of a Point

A flashcard featuring an illustration of a Point

A girl standing with her arm in front of her, pointing at something.

Girl Pointing

A girl standing with her arm in front of her, pointing at something.

"To add 50 yards describe a short horizontal line with forefinger." — Moss, 1914

Hand signal

"To add 50 yards describe a short horizontal line with forefinger." — Moss, 1914

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…

Illustration of a hyperbola with distances to foci drawn. "The difference of the distances of any point from the foci of an hyperbola is greater than or less than 2a, according as the point is on the concave or convex side of the curve."

Foci Distance of Hyperbola

Illustration of a hyperbola with distances to foci drawn. "The difference of the distances of any point…

Illustration of a hyperbola with a line bisecting the focal radii. "If through a point P of an hyperbola a line is drawn bisecting the angle between the focal radii, every point in this line except P is on the convex side of the curve.

Line Bisecting Angle Between Focal Radii in Hyperbola

Illustration of a hyperbola with a line bisecting the focal radii. "If through a point P of an hyperbola…

Illustration of a point on a hyperbola. "If d denotes the abscissa (x-coordinate) of a point of an hyperbola, r and r' its focal radii, then r = ed - a, and r' = ed + a."

Point on a Hyperbola

Illustration of a point on a hyperbola. "If d denotes the abscissa (x-coordinate) of a point of an hyperbola,…

Illustration showing how to draw a tangent to an hyperbola from a given point P on the convex side of the hyperbola.

Tangent to Hyperbola

Illustration showing how to draw a tangent to an hyperbola from a given point P on the convex side of…

Joab's Artifice, from Hans Holbein's series of engravings known as his Bible Cuts.

Joab's Artifice

Joab's Artifice, from Hans Holbein's series of engravings known as his Bible Cuts.

A leaf shaped like a triangle.

Sagittate Leaf

A leaf shaped like a triangle.

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…

"Sectional compartment of the Nave of Lincoln Cathedral." — Encyclopedia Britanica, 1893

Lincoln Cathedral

"Sectional compartment of the Nave of Lincoln Cathedral." — Encyclopedia Britanica, 1893

"Sectional compartment of the Choir of Lincoln Cathedral." — Encyclopedia Britanica, 1893

Lincoln Cathedral

"Sectional compartment of the Choir of Lincoln Cathedral." — Encyclopedia Britanica, 1893

To find the projections of a line which joins two points, A, B given by their projections.

Line Projection

To find the projections of a line which joins two points, A, B given by their projections.

An illustration of a straight line with 5 points dividing it into 4 equal parts. Multiple of a given line.

Straight Line With Points Divided Into Equal Parts

An illustration of a straight line with 5 points dividing it into 4 equal parts. Multiple of a given…

An illustration of a straight line with a point.

Straight Line With Point

An illustration of a straight line with a point.

Illustration of two lines intersecting at a point. This can be used to show vertical angles.

Intersecting Straight Lines

Illustration of two lines intersecting at a point. This can be used to show vertical angles.

Illustration showing two straight lines drawn from the same point in a perpendicular to a given line, cutting off on the line unequal segments from the foot of the perpendicular, the more remote is the greater.

Lines Drawn From the Same Point in a Perpendicular to a Given line, Cutting Off Segments

Illustration showing two straight lines drawn from the same point in a perpendicular to a given line,…

Illustration of two straight lines drawn from a point in a perpendicular to a given line, cutting off on the given line equal segments from the foot of the perpendicular, are equal and make equal angles with the perpendicular. This illustration can be used to show the proof.

Lines Drawn to Another Line to Form Triangle

Illustration of two straight lines drawn from a point in a perpendicular to a given line, cutting off…

An illustration of a man sitting on the shore of a lake and a child standing behind him pointing into to the water.

Man Sitting & Child Standing Near Lake

An illustration of a man sitting on the shore of a lake and a child standing behind him pointing into…

An instrument for making a reduced, enlarged, or exact copy of a plane figure

Pantograph

An instrument for making a reduced, enlarged, or exact copy of a plane figure

Illustration of a parabola showing point P is equidistant to the focus and the directrix.

Point on Parabola

Illustration of a parabola showing point P is equidistant to the focus and the directrix.

Illustration of a parabola. "If a line PT is drawn from any point P of the curve, bisecting the angle between PF and the perpendicular from P to the directrix, every point of the line PT, except P, is without the curve."

Point on Parabola

Illustration of a parabola. "If a line PT is drawn from any point P of the curve, bisecting the angle…

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.

To draw a plane through a given point parallel to a given plane.

Parallel Planes

To draw a plane through a given point parallel to a given plane.

Two planes lie perpendicular to one another. A line perpendicular to plane 1 and a line perpendicular to plane 2 will meet at a point, A, and form a perpendicular intersection.

Perpendicular Planes

Two planes lie perpendicular to one another. A line perpendicular to plane 1 and a line perpendicular…

"Potent counter-potent, sometimes called varry cuppy, differs from potent in that the potents of the same tincture are placed base to base and point to point."—Aveling, 1891

Potent Counter-Potent Shield

"Potent counter-potent, sometimes called varry cuppy, differs from potent in that the potents of the…

Given any three circles, the common chords meet at one point.

Radical Center of 3 Circles

Given any three circles, the common chords meet at one point.

"Diagrammatic section through the eyeball. xx, optic axis; k, nodal point." —Martin, 1917

Retina

"Diagrammatic section through the eyeball. xx, optic axis; k, nodal point." —Martin, 1917

"Diagram illustrating the points at which incident rays meet the retina. xx, optic axis; k, first nodal point; k', second nodal point; b, point where the image of B would be formed, were the eye properly accommodated for it; a, the retinal point where the image of A would be formed." —Martin, 1917

Retina

"Diagram illustrating the points at which incident rays meet the retina. xx, optic axis; k, first nodal…

Round head and flat head screws compared.

Wood Screws

Round head and flat head screws compared.

Skadi is a giantess and daughter of Thiassi who was killed while chasing Loki because of a trick. Skadi asks the gods for something in return for the loss of her father. They let her choose a husband by looking at only the gods' feet. She chooses a pair of white, slim feet which belong to Niord, the wind god.

Skadi Chooses a Husband

Skadi is a giantess and daughter of Thiassi who was killed while chasing Loki because of a trick. Skadi…