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Illustration showing the golden angle. The golden angle is the smaller of two angles created by dividing the circumference of a circle according to the golden section. The ratio of the length of the larger arc to the smaller arc is equal to the ratio of the entire circumference to the larger arc. The golden angle is approximately 137.51°.

Golden Angle

Illustration showing the golden angle. The golden angle is the smaller of two angles created by dividing…

Illustration of 4 congruent rectangular prisms placed in the shape of a square. They are arranged to look like they are 3-dimensional rectangular solids coming out of the page.

4 Congruent Rectangular Prisms

Illustration of 4 congruent rectangular prisms placed in the shape of a square. They are arranged to…

Illustration of a cyclic pentagon, a pentagon inscribed in a circle. This can also be described as a circle circumscribed about a pentagon. In this illustration, the pentagon is not regular (the lengths of the sides are not equal).

Cyclic Pentagon

Illustration of a cyclic pentagon, a pentagon inscribed in a circle. This can also be described as a…

Illustration of a cyclic quadrilateral, a quadrilateral inscribed in a circle. This can also be described as a circle circumscribed about a quadrilateral. In this illustration, the quadrilateral is not regular (the lengths of the sides are not equal).

Cyclic Quadrilateral

Illustration of a cyclic quadrilateral, a quadrilateral inscribed in a circle. This can also be described…

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle circumscribed about a hexagon. In this illustration, the hexagon is not regular (the lengths of the sides are not equal).

Cyclic Hexagon

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle…

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle circumscribed about a hexagon. In this illustration, the hexagon is not regular (the lengths of the sides are not equal).

Cyclic Hexagon

Illustration of a cyclic hexagon, a hexagon inscribed in a circle. This can also be described as a circle…

Illustration used to prove the Pythagorean Theorem, according to Euclid. A perpendicular is drawn from the top vertex of the right triangle extended through the bottom square, forming 2 rectangles. Each rectangle has the same area as one of the two legs. This proves that the sum of the squares of the legs is equal to the square of the hypotenuse (Pythagorean Theorem).

Euclid's Pythagorean Theorem Proof

Illustration used to prove the Pythagorean Theorem, according to Euclid. A perpendicular is drawn from…

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is equal to the square of the hypotenuse.

Geometric Pythagorean Theorem Proof

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is…

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. The geometrical illustration depicts a 3,4,5 right triangle with the square units drawn to prove that the sum of the squares of the legs (9 + 16) equals the square of the hypotenuse.

Geometric Pythagorean Theorem Proof

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is…

A visual illustration used to prove the Pythagorean Theorem by rearrangement. When the 4 identical triangles are removed, the areas are equal. Thus, proving the sum of the squares of the legs is equal to the square of the hypotenuse.

Pythagorean Theorem Proof by Rearrangement

A visual illustration used to prove the Pythagorean Theorem by rearrangement. When the 4 identical triangles…

Leaves - simple; indeterminate in position because of their smallness and closeness. They are arranged in four rows up and down the branchlets. In younger or rapidly growing sprouts the leaves are awl-shaped or needle-shaped, somewhat spreading from the branch, very sharp and stiff, placed in pairs (or sometimes in threes), usually about one fourth of an inch long, and with the fine branchlets, which they cover, rounded. In the older and slower-growing trees the leaves are scale-like and overlapping, egg-shape, closely pressed to the branchlets which they cover, and with the branchlets square. As the branchlets grow, the lower scales sometimes lengthen and become dry and chaffy and slightly spreading. Bark - brown and sometimes purplish-tinged, often shredding off with age and leaving the trunk smooth and polished. Berries - about the size of a small pea, closely placed along the branchlets, bluish, and covered with a whitish powder. Found - in Southern Canada, and distributed nearly throughout the United States - more widely than any other of the cone-bearing trees. General information - An evergreen tree, fifteen to thirty feet high (much larger at the South), usually pyramid-shaped, with a rounded base, but varying very greatly, especially near the coast, where it is often twisted and flattened into angular and weird forms. The wood is very valuable, light, straight-grained, durable, fragrant. It is largely used for posts, for cabinet-work, for interior finish, and almost exclusively in the making of lead pencils. The heart-wood is usually a dull red (whence the name), the sap-wood white. Among the most picturesque objects in the Turkish landscape, standing like sentinels, singly or in groups, and slender and upright as a Lombardy Poplar, are the black cypress trees (C. sempervirens). They mark the sites of graves, often of those which have long since disappeared. In America, more than any other northern tree, the red cedar gives the same sombre effect, whether growing wild or planted in cemeteries. The Common Juniper (J. communis, L.), common as a shrub, is occasionally found in tree form, low, with spreading or drooping branches, and with leaves resembling those of a young Red Cedar, awl-shaped and spreading, but arranged in threes instead of opposite.

Genus Juniperus, L. (Red Cedar)

Leaves - simple; indeterminate in position because of their smallness and closeness. They are arranged…

For larger circles beam compasses are used. The two parts called channels which carry the pen and the needle point are clamped to a wooden beam at a distance equal to the radius of the circle. The thumb nut underneath one of the channel pieces makes accurate adjustment possible.

Beam Compasses

For larger circles beam compasses are used. The two parts called channels which carry the pen and the…

Diamond pattern exercise: Divide A D and A B into 4 equal parts, then draw horizontal lines through E, F, and G and vertical lines through L, M, and N. Draw lines from A and B to the intersection O of lines E and M, and from A and D to the intersection P of lines F and L. Similarly, draw D J, J C, C I, and I B. Also connect the points O, P, J, and I, thus forming a square. The four diamond-shaped areas are formed by drawing lines from the middle points of A D, A B, B C, and D C to the middle points of lines A P, A O, O B, I B, etc.

Diamond Exercise

Diamond pattern exercise: Divide A D and A B into 4 equal parts, then draw horizontal lines through…

Square pattern exercise: Divide A D and A B into 4 equal parts and draw horizontal and vertical lines. Now divide these dimensions, A L, M N, etc., and E F, G B, etc., into 4 equal parts- each 1/4 inch- and draw light pencil lines with the T-square and triangle.

Square Exercise

Square pattern exercise: Divide A D and A B into 4 equal parts and draw horizontal and vertical lines.…

Weave pattern exercise: Divide A D and A B into 8 equal parts, and through the points O, P, Q, H, I, J, etc., draw horizontal and vertical lines. Now draw lines connecting O and H, P and I, Q and J, etc. As these lines form an angle of 45 degrees with the horizontal, a 45-degree triangle may be used. Similarly from each one of the given points on A B and A D, draw lines at an angle of 45 degrees to B C and D C respectively.

Weave Exercise

Weave pattern exercise: Divide A D and A B into 8 equal parts, and through the points O, P, Q, H, I,…

Five students are learning how to use scales to measure a book. They have balanced the book with some weights.

Children Using Scales

Five students are learning how to use scales to measure a book. They have balanced the book with some…

(1811-1884) American orator and reformer who led the anti-slavery campaign and fought for the rights of women and Native Americans.

Wendell Phillips

(1811-1884) American orator and reformer who led the anti-slavery campaign and fought for the rights…

A poisonous plant that climbs and has groups of three leaflets.

Poison Ivy

A poisonous plant that climbs and has groups of three leaflets.

This figure "shows the cotidal lines and the lines of equal rise and fall for a diurnal component in latitude 30 degrees north." -Coast and Geodetic Survey, 1901

Cotidal Lines

This figure "shows the cotidal lines and the lines of equal rise and fall for a diurnal component in…

This figure shows "the cotidal lines and lines of equal amplitude for a diurnal tide in a circular sea of 20 degrees radius, the latitude of the center being 30 degrees north." -Coast and Geodetic Survey, 1901

Diurnal Cotidal Lines

This figure shows "the cotidal lines and lines of equal amplitude for a diurnal tide in a circular sea…

This figure shows "the cotidal lines and lines of equal amplitude for a diurnal tide in a circular sea of 20 degrees radius, the latitude of the center being 30 degrees north." -Coast and Geodetic Survey, 1901

Diurnal Cotidal Lines

This figure shows "the cotidal lines and lines of equal amplitude for a diurnal tide in a circular sea…

The figure shows the harmony and syzygy of tides in four different places (the four corners of the square).

Tide Theory, Square

The figure shows the harmony and syzygy of tides in four different places (the four corners of the square).

The figure shows the harmony and syzygy of tides in four different places (the four corners of the square).

Tide Theory, Square

The figure shows the harmony and syzygy of tides in four different places (the four corners of the square).

The figure shows a 30 degree triangle made from a square depicting the harmony and syzygy of tides in four different places (the four corners of the square).

Tide Theory, Triangle

The figure shows a 30 degree triangle made from a square depicting the harmony and syzygy of tides in…

The figure shows a 30 degree triangle made from a square depicting the harmony and syzygy of tides in four different places (the four corners of the square).

Tide Theory, Triangle

The figure shows a 30 degree triangle made from a square depicting the harmony and syzygy of tides in…

"The line AB represents the surface of a water wave whose length is ai, moving in the direction of the long arrow. Suppose the wave length to be divided into eight equal parts, ab, bc, etc. If the water were still, the particles 1, 2, 3, etc., would lie directly above the points a, b, c, etc., but each particle is moving in a circular path in the direction shown by the curved arrows." -Dryer, 1901

Wave Movement

"The line AB represents the surface of a water wave whose length is ai, moving in the direction of the…

Illustration used to prove that all right angles are equal.

Equal Right Angles

Illustration used to prove that all right angles are equal.

Illustration showing two congruent pentagons.

Congruent Pentagons

Illustration showing two congruent pentagons.

Illustration showing two parallel vertical lines cut by a perpendicular line and a transversal. Congruent line segments are marked.

Parallel Lines Cut By a Perpendicular And Transversal

Illustration showing two parallel vertical lines cut by a perpendicular line and a transversal. Congruent…

Illustration showing that if two angles of a triangle are equal, the bisectors of these angles are equal.

Angle Bisectors In An Isosceles Triangle

Illustration showing that if two angles of a triangle are equal, the bisectors of these angles are equal.

Illustration showing that if equal segments measured from the vertex are laid off on the arms of an isosceles triangle, the lines joining the ends of these segments to the opposite ends of the base will be equal.

Equal Segments In An Isosceles Triangle

Illustration showing that if equal segments measured from the vertex are laid off on the arms of an…

Illustration showing that if equal segments prolonged through the vertex are laid off on the arms of an isosceles triangle, the lines joining the ends of these segments to the opposite ends of the base will be equal.

Equal Segments In An Isosceles Triangle

Illustration showing that if equal segments prolonged through the vertex are laid off on the arms of…

Illustration showing that can be used to prove that the base angles of an isosceles triangle are equal.

Base Angles In An Isosceles Triangle

Illustration showing that can be used to prove that the base angles of an isosceles triangle are equal.

Illustration showing that if equal segments measured from the end of the base are laid off on the base of an isosceles triangle, the lines joining the vertex of the triangle to the ends of the segments will be equal.

Equal Segments In An Isosceles Triangle

Illustration showing that if equal segments measured from the end of the base are laid off on the base…

Illustration showing that if equal segments measured from the end of the base prolonged are laid off on the base of an isosceles triangle, the lines joining the vertex of the triangle to the ends of the segments will be equal.

Equal Segments In An Isosceles Triangle

Illustration showing that if equal segments measured from the end of the base prolonged are laid off…

Illustration used to prove that triangle EFD is equilateral given that triangle ABC is equilateral and AE=BF=CD.

Equilateral Triangle Inscribed In An Equilateral Triangle

Illustration used to prove that triangle EFD is equilateral given that triangle ABC is equilateral and…

Illustration used to show that two triangles are equal if the three sides of one are equal respectively to the three sides of the other.

Equal Triangles

Illustration used to show that two triangles are equal if the three sides of one are equal respectively…

Illustration used to show how to construct an equilateral triangle, with a given line as a side.

Construction Of Equilateral Triangle

Illustration used to show how to construct an equilateral triangle, with a given line as a side.

Illustration used to show how to construct an angle equal to a given angle when given a vertex and a given side.

Construction Of An Equal Angle

Illustration used to show how to construct an angle equal to a given angle when given a vertex and a…

"Maximum extinction angles in the Pyroxene and Amphibole groups. Solid lines indicate extinction angles from c to c; broken lines from c to a. The extinction angle in an amphibole is generally less than 23 degrees; in a pyroxene it is generally greater. " -Johannsen, 1908

Extinction Angles

"Maximum extinction angles in the Pyroxene and Amphibole groups. Solid lines indicate extinction angles…

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second."

2 Triangles Theorem

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides…

Illustration of triangle ABC with BE extended through the triangle at point D. Segment AB is equal to segment BD.

Segments Labeled In A Triangle

Illustration of triangle ABC with BE extended through the triangle at point D. Segment AB is equal to…

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides of the other, but the third side of the first greater than the third side of the second, then the angle opposite the third side of the first is greater than the angle opposite the third side of the second."

2 Triangles Theorem

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides…

Illustration used to prove the theorem, "If two straight lines are cut by a transversal making a pair of alternate interior angles equal, the lines are parallel."

Parallel Lines Cut By A Transversal Theorem

Illustration used to prove the theorem, "If two straight lines are cut by a transversal making a pair…

Illustration used to prove the theorem, "If two parallel lines are cut by a transversal, the alternate interior angles are equal."

Parallel Lines Cut By A Transversal Theorem

Illustration used to prove the theorem, "If two parallel lines are cut by a transversal, the alternate…

Illustration used to prove the theorem, "Two angles whose sides are parallel, each to each, are either equal or supplementary."

2 Angles With Parallel Sides Theorem

Illustration used to prove the theorem, "Two angles whose sides are parallel, each to each, are either…

Illustration used to prove the theorem, "Two angles whose sides are perpendicular, each to each, are either equal or supplementary."

2 Angles With Perpendicular Sides Theorem

Illustration used to prove the theorem, "Two angles whose sides are perpendicular, each to each, are…

An illustration of a circle inscribed in a square. It can be used to show that the area of a circle is .7854 of the area of a square whose sides are equal to its diameter.

Circle Inscribed In A Square

An illustration of a circle inscribed in a square. It can be used to show that the area of a circle…

"Elliptic.--Nearly oval, but of equal breadth at each end." -Newman, 1850

Elliptical Leaf

"Elliptic.--Nearly oval, but of equal breadth at each end." -Newman, 1850

In the early developments of the races of mankind often groups were named based on common characteristics. Bushwomen could be found in the Mediterranean.

Mediterranean Bushwoman

In the early developments of the races of mankind often groups were named based on common characteristics.…

"Accessory buds of box-elder (Negundo). A, front view of group; B, two groups seen in profile." -Bergen, 1896

Maple Buds

"Accessory buds of box-elder (Negundo). A, front view of group; B, two groups seen in profile." -Bergen,…

An illustration showing the construction used to divide a line AB into two equal parts; and to erect a perpendicular through the middle. "With the end A and B as centers, draw the dotted circle arcs with a radius greater than half the line. Through the crossings of the arcs draw the perpendicular CD, which divides the line into two equal parts."

Construction Of A Line Divided In Equal Parts

An illustration showing the construction used to divide a line AB into two equal parts; and to erect…

An illustration showing the construction used to erect a perpendicular. "With C as a center, draw the dotted circle arcs at A and B equal distances from C. With A and B as centers, draw the dotted circle arcs at D. From the crossing D draw the required perpendicular DC."

Construction Of A Perpendicular

An illustration showing the construction used to erect a perpendicular. "With C as a center, draw the…

An illustration showing the construction used to erect a perpendicular from a point to a line. "With C as a center, draw the dotted circle arc so that it cuts the line at A and B. With A and B as centers, draw the dotted cross arcs at D with equal radii. Draw the required perpendicular through C and crossing D."

Construction Of A Perpendicular

An illustration showing the construction used to erect a perpendicular from a point to a line. "With…

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…

An illustration showing the construction used to divide an angle into two equal parts. "With C as a center, draw the dotted arc DE; with D and E as centers, draw the cross arcs at F with equal radii. Join CF, which divides the angle into the required parts."

Construction Of A Divided Angle

An illustration showing the construction used to divide an angle into two equal parts. "With C as a…

An illustration showing the construction used to divide an angle into two equal parts when the lines do not extend to a meeting point. "Draw the lined CD and CE parallel, and at equal distances from the lines AB and FG. With C as a center, draw the dotted arc BG; and with B and G as centers, draw the cross arcs H. Join CD, which divides the angle into the required equal parts."

Construction Of A Divided Angle

An illustration showing the construction used to divide an angle into two equal parts when the lines…

An illustration showing how to construct a square upon a given line. "With AB as radius and A and B as centers, draw the circle arcs AED and BEC. Divide the arc BE in two equal parts at F, and with EF as radius and E as center, draw the circle CFD. Join A and CB and D, C and D, which completes the required square."

Square Constructed Upon A Given Line

An illustration showing how to construct a square upon a given line. "With AB as radius and A and B…

"In the single fixed pulley (fig. 1) there is no mechanical advantage, the power and weight being equal. It may be considered as a lever of the first kind with equal arms. In the single movable pulley (fig. 2) where the cords are parallel there is a mechanical advantage, there being an equilibrium when the power is considered as a lever of the second kind, in which the distance of the power from the fulcrum is double hat of the weight from the fulcrum." -Marshall

Pulley

"In the single fixed pulley (fig. 1) there is no mechanical advantage, the power and weight being equal.…

An illustration showing how to construct a cyma, or two circle arcs that will tangent themselves, and two parallel lines at given points A and B. "Join A and B; divide AB into four equal parts and erect perpendiculars. Draw Am at right angles from A, and Bn at right angles from B; then m and n are the centers of the circle arcs of the required cyma."

Construction Of A Cyma

An illustration showing how to construct a cyma, or two circle arcs that will tangent themselves, and…