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Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lengths of lines a and b are the same. You would need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lengths of lines a and b are the same. You would need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lines are the same distance apart. You may need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lines are the same distance apart. You may need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lines are prolongations of the other lines. You may need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lines are prolongations of the other lines. You may need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…

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 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 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…

"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…

"Simple orthoclase crystal showing cleavage lines and the position of the twinning plane in a Carlsbad twin." -Johannsen, 1908

Orthoclase Crystal

"Simple orthoclase crystal showing cleavage lines and the position of the twinning plane in a Carlsbad…

Illustration used to prove that "If two straight lines are parallel to a third straight line, they are parallel to each other."

Parallel Lines Theorem

Illustration used to prove that "If two straight lines are parallel to a third straight line, they are…

Illustration of two straight lines cut by a transversal. The 8 angles formed are labeled.

2 Lines Cut By A Transversal

Illustration of two straight lines cut by a transversal. The 8 angles formed are labeled.

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 of two straight lines, m and n, cut by a transversal t.

2 Lines Cut By A Transversal

Illustration of two straight lines, m and n, cut by a transversal t.

Illustration of two straight lines, AB and CD, cut by a transversal EF.

2 Lines Cut By A Transversal

Illustration of two straight lines, AB and CD, cut by a transversal EF.

Illustration of the construction used to create a line parallel to a given line.

Construction Of A Parallel Line

Illustration of the construction used to create a line parallel to a given line.

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 corollary that "Two lines perpendicular respectively to two intersecting lines also intersect."

Intersecting Lines Corollary

Illustration used to prove the corollary that "Two lines perpendicular respectively to two intersecting…

"Section of the shell of Nautilus pompilius, showing the septa (s, s), the septal necks (s. n., s. n.), the siphuncle (si, represented by dotted lines), and the large body-chamber (ch)." -Parker, 1900

Chambered Nautilus

"Section of the shell of Nautilus pompilius, showing the septa (s, s), the septal necks (s. n., s. n.),…

"Spirula peronii, lateral view. d, terminal sucker; f, funnel; s1, s2, projecting portions of the shell, the internal part of which is indicated by dotted lines." -Parker, 1900

Spirula Peronii

"Spirula peronii, lateral view. d, terminal sucker; f, funnel; s1, s2, projecting portions of the shell,…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes labeled and increments from -5 to 5.

-5 To 5 Coordinate Grid With Axes Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments from -5 to 5 on each axis. Axes and increments are not labeled.

-5 To 5 Coordinate Grid With Grid Lines Shown, But No Labels

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes and increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Increments And Axes Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes and odd increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Axes And Odd Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with odd increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Odd Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with odd…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the axes and some increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Axes And Some Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with some increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Some Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with some…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the axes and some increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Axes And Some Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with some increments at -5 to 5 labeled.

-5 To 5 Coordinate Grid With Some Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with some…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes and even increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Axes And Even Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with even increments from -5 to 5 labeled.

-5 To 5 Coordinate Grid With Even Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with even…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes labeled and increments from -10 to 10.

-10 To 10 Coordinate Grid With Axes Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments from -10 To 10 on each axis. Axes and increments are not labeled.

-10 To 10 Coordinate Grid With Grid Lines Shown, But No Labels

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes and increments from -10 To 10 labeled.

-10 To 10 Coordinate Grid With Increments And Axes Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments from -10 To 10 labeled.

-10 To 10 Coordinate Grid With Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes and some increments from -10 To 10 labeled.

-10 To 10 Coordinate Grid With Axes And Some Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with some increments from -10 To 10 labeled.

-10 To 10 Coordinate Grid With Some Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with some…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes and increments from -10 To 10 labeled by 5s.

-10 To 10 Coordinate Grid With Axes And Increments Labeled By 5s And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments from -10 To 10 labeled by 5s.

-10 To 10 Coordinate Grid With Increments Labeled By 5s And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with increments…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with axes and some increments from -10 To 10 labeled.

-10 To 10 Coordinate Grid With Axes And Increments Labeled By 10s And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with axes…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with some increments from -10 To 10 labeled.

-10 To 10 Coordinate Grid With Increments Labeled By 10s And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with some…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the x- and y-axes and even increments from -10 to 10 labeled.

-10 To 10 Coordinate Grid With Axes And Even Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with the…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with even increments from -10 to 10 labeled.

-10 To 10 Coordinate Grid With Even Increments Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with even…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system. Neither axis is labeled.

Blank Coordinate Grid With Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system. Neither…

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with both axes labeled.

Coordinate Grid With Axes Labeled And Grid Lines Shown

Illustration of an xy grid/graph with grid lines shown. It is the Cartesian coordinate system with both…

An illustration showing the construction used to erect a parallel line. "With C as a center, draw the dotted arc ED, with E as a center, draw through C the dotted arc F.C. With the radius FC and E as a center, draw the cross arc at D. Join C with the cross at D, which will be the required parallel line.

Construction Of A Parallel

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

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 the construction used to erect a parallelogram given two sides and an angle. "Draw the base line DE, and make the angle FDE = C; lines DE = B and DF = A; complete the parallelogram by cross arcs at G, and the problem is thus solved."

Construction Of A Parallelogram

An illustration showing the construction used to erect a parallelogram given two sides and an angle.…

An illustration showing the construction used to divide the line AB in the same proportion of parts as AC. "Join C and B, and through the given divisions 1, 2, and 3 draw lines parallel with CB, which solves the problem."

Divide A Line Proportionately

An illustration showing the construction used to divide the line AB in the same proportion of parts…

An illustration showing how to construct two circles that tangent themselves and two given lines. "Draw the center line AB between the given lines; assume D to be the tangenting point of the circles; draw DC at right angles to AB. With C as center and CD as radius, draw the circle EDF. From E, draw Em at right angles to EF; and from F draw Fm at right angles to FE; then m and n are the centers for the required circles."

Construction Of Two Circles That Tangent Themselves and 2 Given Lines

An illustration showing how to construct two circles that tangent themselves and two given lines. "Draw…

An illustration showing how to construct a circle that tangents two given lines inclined to one another with the one tangenting point being given. "Draw the center line GF. From E, draw EF at right angles to AB; then F is the center of the circle required.

Construction Of A Circle That Tangents 2 Given Lines

An illustration showing how to construct a circle that tangents two given lines inclined to one another…

An illustration showing how to construct a circle that tangents two given lines and goes through a given point c on the line FC, which bisects the angle of the lines. "Through C draw AB at right angles to CF; bisect the angles DAB and EBA, and the crossing on CF is the center of the required circle."

Construction Of A Circle That Tangents 2 Given Lines And Goes Through A Given Point

An illustration showing how to construct a circle that tangents two given lines and goes through a given…

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…

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

Construction Of A Talon

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

An illustration showing how to construct a circle arc without recourse to its center, but its chord AB and height h being given. "With the chord as radius, and A and B as centers, draw the dotted circle arcs AC and BD. Through the point O draw the lines AOo and BOo. Make the arcs Co=Ao and Do=Bo. Divide these arcs into any desired number of equal parts, and number them as shown on the illustration. Join A and B with the divisions, and the crossings of equal numbers are points in the circle arc."

Construction Of A Circle Arc

An illustration showing how to construct a circle arc without recourse to its center, but its chord…

In field fortification, the simplest kind of work employed, consisting of two parapets of earth raised so as to form a salient angle, with the apex towards the enemy and unprotected on the rear. Several redans connected by curtains form lines of intrenchment.

Redans

In field fortification, the simplest kind of work employed, consisting of two parapets of earth raised…