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Projections of a line inclined to a plane.

Projections of a Line

Projections of a line inclined to a plane.

Projections of a line inclined to both the horizontal and vertical planes will not show its true length, as represented in this drawing.

Projections of a Line

Projections of a line inclined to both the horizontal and vertical planes will not show its true length,…

Projections of a line's true length by revolving horizontal projection.

Projections of a Line

Projections of a line's true length by revolving horizontal projection.

Projections of a line's true length by revolving vertical projection.

Projections of a Line

Projections of a line's true length by revolving vertical projection.

Illustration showing how a rolling logarithmic spiral can be constructed by taking successive values of theta, starting from oc. The values of r can be calculated and then the curve can be plotted.

Construction of Logarithmic Spiral

Illustration showing how a rolling logarithmic spiral can be constructed by taking successive values…

Illustration showing the rolling of two logarithmic spirals of equal obliquity.

Rolling of Logarithmic Spirals

Illustration showing the rolling of two logarithmic spirals of equal obliquity.

"Orthorhombic. Crystals commonly tabular parallel to basal plane, showing also short prisms and low brachydomes. The brachydomes usually striated parallel to the brachy-axis. Often twinned, given coxcomb and spear-shaped groups." — Ford, 1912

Marcasite

"Orthorhombic. Crystals commonly tabular parallel to basal plane, showing also short prisms and low…

"Suppose the mirror, a b, to be placed on the side of a room, and a lamp to be set in antoher room, but so situated as that its light would shine upon the glass. The lamp itself could not be seen by the eye placed at e, because the partition d is between them; but its image would be visible at e, beacuse the angle of the incident ray, coming from the light, and that of the reflected ray which reaches the eye, are equal." -Comstock 1850

Plane Mirror

"Suppose the mirror, a b, to be placed on the side of a room, and a lamp to be set in antoher room,…

"Suppose the arrow a to be the object reflected b the mirror d c; the incident rays a, flowing from the end of the arror being thrown back by reflection, will meet the eye in the same state of dicergence that they would do, if they proceded to the same distance behind the mirrr, that the eye is before it, as at o." -Comstock 1850

Plane Mirror

"Suppose the arrow a to be the object reflected b the mirror d c; the incident rays a, flowing from…

"If an object is placed before a plane mirror, a virtual image appears behind the mirror. Each point of this image seems to be as far behind the mirror as the corresponding point of the object is in front of the mirror. Hence, images seen in still, clear, water are inverted." -Avery 1895

Plane Mirror

"If an object is placed before a plane mirror, a virtual image appears behind the mirror. Each point…

"A plane mirror reflects from a flat surface, like a common looking glass." —Quackenbos 1859

Plane Mirror

"A plane mirror reflects from a flat surface, like a common looking glass." —Quackenbos 1859

"By placing two plane mirrors facing each other, we may produce an indefinite series of images of an object between them. Each image acts as a material object with respect to the other mirror, in which we see an image of the first image, etc. When the mirrors are placed so as to form with each other an angle that is an aliquot part of 360 degrees, the number of images is one less than the quotient obtained by dividing four right angles by the included angle, provuded that quotient is an even number." -Avery 1895

Multiple Images in a Plane Mirror

"By placing two plane mirrors facing each other, we may produce an indefinite series of images of an…

"The construction for the image produced by a plane mirror depends upon the fact that the image of the object may be located by locating the images of a number of well chosen points in the surface of the object." -Avery 1895

Object in Plane Mirror

"The construction for the image produced by a plane mirror depends upon the fact that the image of the…

"The incident rays, a and b, being parallel before they reach the reflectors, are thrown off at unequal angles in respect to each other, for b fallson the mirror more obliquely than a, and consequently is thrown off more obliquely in a contraty direction, therefore, then angles of reflection being equal to those of incidence, the two rays meet at c." -Comstock 1850

Plane Inclined Mirrors

"The incident rays, a and b, being parallel before they reach the reflectors, are thrown off at unequal…

An airplane with a single wing

Early Design Monoplane

An airplane with a single wing

"The moulding plane." — Encyclopedia Britannica, 1893

Moulding Plane

"The moulding plane." — Encyclopedia Britannica, 1893

Illustration showing that the rolling of non-cylindrical surfaces. "If the angular velocity ratio of two rolling bodies is not a constant, the pitch lines take, the conditions of pure rolling contact should be fulfilled, namely, the point of contact must be on the line of centres, and the rolling arcs must be of equal length.

Rolling of Non-cylindrical Surfaces

Illustration showing that the rolling of non-cylindrical surfaces. "If the angular velocity ratio of…

Illustration of a hollow nonagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid.

Frustum Of A Nonagonal Pyramid

Illustration of a hollow nonagonal pyramid that has been cut by a plane parallel to the base. The top…

"Two stars in the Big Dipper opposite the handle indicate the direction toward the North Star, which, though not very bright, is the first very noticeable star in line." -Wiswell, 1913

Finding the North Star

"Two stars in the Big Dipper opposite the handle indicate the direction toward the North Star, which,…

Illustration of a right octagonal pyramid, with hidden edges shown, that has been cut by a plane parallel to the base. The lower part is known as the frustum.

Octagonal Pyramid Cut By Plane

Illustration of a right octagonal pyramid, with hidden edges shown, that has been cut by a plane parallel…

Illustration of a right octagonal pyramid that has been cut by a plane parallel to the base. The lower part is known as the frustum.

Octagonal Pyramid Cut By Plane

Illustration of a right octagonal pyramid that has been cut by a plane parallel to the base. The lower…

Illustration of a hollow octagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid.

Frustum Of A Octagonal Pyramid

Illustration of a hollow octagonal pyramid that has been cut by a plane parallel to the base. The top…

"A contact twin, since the two individuals lie simply in contact with each other upon a certain plane." — Ford, 1912

Twinned octahedron

"A contact twin, since the two individuals lie simply in contact with each other upon a certain plane."…

An orthographic projection of a pyramid from three dimensional space into two dimensional space

Orthographic Pyramid Projection

An orthographic projection of a pyramid from three dimensional space into two dimensional space

Diagram of a cone intersected by plane J to form a parabola. Also pictured is a circle formed by the intersection of plane K with the cone. "Every point of a parabola is equidistant from the focus and the directrix."

Cone Intersected by a Plane to Form a Parabola

Diagram of a cone intersected by plane J to form a parabola. Also pictured is a circle formed by the…

Diagram showing how to construct a parabola when given the directrix (d), the focus (F) and the eccentricity (m).

Construction of a Parabola

Diagram showing how to construct a parabola when given the directrix (d), the focus (F) and the eccentricity…

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.

Illustration of a right pentagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Pentagonal Pyramid

Illustration of a right pentagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a hollow pentagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid.

Frustum Of A Pentagonal Pyramid

Illustration of a hollow pentagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a right pentagonal pyramid, with a height much larger than the edge lengths of the base, that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Pentagonal Pyramid

Illustration of a right pentagonal pyramid, with a height much larger than the edge lengths of the base,…

Illustration of a right pentagonal pyramid, with a height smaller than the edge lengths of the base, that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Pentagonal Pyramid

Illustration of a right pentagonal pyramid, with a height smaller than the edge lengths of the base,…

Illustration of a hollow pentagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Pentagonal Pyramid

Illustration of a hollow pentagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a skewed pentagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Skewed Pentagonal Pyramid

Illustration of a skewed pentagonal pyramid that has been cut by a plane parallel to the base. The top…

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…

A pusher aeroplane plan illustrating the proper ribbing and wiring for the plane. The wires are used to adjust the wings during flight. Pusher aeroplane propeller is located behind the cockpit.

Front View Pusher Aeroplane Plan

A pusher aeroplane plan illustrating the proper ribbing and wiring for the plane. The wires are used…

A propeller airplane plan illustrating the proper location of the skeleton and placement of the wires controlling the adjustable wings.

Propeller Aeroplane Side Plan

A propeller airplane plan illustrating the proper location of the skeleton and placement of the wires…

A top view plan of a propeller aeroplane, or airplane. The plan shows the proper ribs and skeleton for the wings and back side of the plane.

Propeller Aeroplane Top Plan

A top view plan of a propeller aeroplane, or airplane. The plan shows the proper ribs and skeleton for…

A side view plan of a propeller aeroplane for two people. The plan illustrates the the proper location of the wires to support the the wings on the plane.

Propeller Aeroplane Top Plan

A side view plan of a propeller aeroplane for two people. The plan illustrates the the proper location…

A plan for pusher aeroplane, or airplanes. The propeller of the pusher plane is located behind the cockpit to "push" the plane while flying. The plan illustrates the proper skeleton and wires to control the adjustable wings.

Pusher Aeroplane Plan

A plan for pusher aeroplane, or airplanes. The propeller of the pusher plane is located behind the cockpit…

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…

A plane is a tool for shaping wood. Planes are used to flatten, reduce the thickness of, and impart a smooth surface to a rough piece of lumber or timber. Planing is used to produce horizontal, vertical, or inclined flat surfaces on workpieces usually too large for shaping. Special types of planes are designed to cut joints or decorative mouldings.

Plane Tool

A plane is a tool for shaping wood. Planes are used to flatten, reduce the thickness of, and impart…

An inclined plane used to multiply work done.

Inclined Plane

An inclined plane used to multiply work done.

Illustration of a plane determined by 2 intersecting lines

Plane Determined by 2 Intersecting Lines

Illustration of a plane determined by 2 intersecting lines

Illustration of a straight line projected upon a plane.

Line Projected Upon Plane

Illustration of a straight line projected upon a plane.

"The locus of a point in space equidistant from the extremities of a straight line is the plane perpendicular to this line at its middle point."

Locus Point to a Plane

"The locus of a point in space equidistant from the extremities of a straight line is the plane perpendicular…

Oblique lines drawn from a point to a plane.

Oblique Lines Drawn to a Plane

Oblique lines drawn from a point to a plane.

Illustration of a plane determined by 2 parallel lines

Plane Determined by 2 Parallel Lines

Illustration of a plane determined by 2 parallel lines

If a straight line is perpendicular to each of two other straight lines in a plane at their point of intersection, it is perpendicular to the plane of the two lines.

Line Perpendicular to Plane

If a straight line is perpendicular to each of two other straight lines in a plane at their point of…

Straight line AB is perpendicular to the lines in plane MN at point B.

Line Perpendicular to Plane

Straight line AB is perpendicular to the lines in plane MN at point B.

Straight line AB is perpendicular to the lines in plane MN at point B. "If from the foot of a perpendicular to a plane a straight line is drawn at right angles to any line in the plane, the line drawn from its intersection with the line in the plane to any point of the perpendicular is perpendicular to the line of the plane."

Line Perpendicular to Plane

Straight line AB is perpendicular to the lines in plane MN at point B. "If from the foot of a perpendicular…

Two straight lines, AB and CD, perpendicular to the same plane (MN) are parallel.

Lines Perpendicular to Plane

Two straight lines, AB and CD, perpendicular to the same plane (MN) are parallel.

Diagram used to prove the theorem: "If a pyramid is cut by a plane parallel to the base, the edges are divided proportionally, and the section is a polygon similar to the base."

Pyramid Cut By Plane

Diagram used to prove the theorem: "If a pyramid is cut by a plane parallel to the base, the edges are…

Illustration of a plane rotated about a straight line.

Plane Rotated About a Straight Line

Illustration of a plane rotated about a straight line.

Diagram used to prove the theorem: "Every section of a sphere by a lane is a circle."

Sphere Cut By Plane

Diagram used to prove the theorem: "Every section of a sphere by a lane is a circle."

Diagram used to prove the theorem: "A plane perpendicular to a radius at its extremity is tangent to the sphere."

Sphere Tangent to Plane

Diagram used to prove the theorem: "A plane perpendicular to a radius at its extremity is tangent to…

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through its center, and a similar point is found on the line at an equal distance beyond the center." — Ford, 1912

Symmetry plane

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through…

Illustration of three intersecting planes. "The intersections of two parallel planes by a third plane are parallel lines."

3 Intersecting Planes

Illustration of three intersecting planes. "The intersections of two parallel planes by a third plane…

Illustration of three intersecting planes. "The straight line perpendicular to one of two parallel lanes is perpendicular to the other."

3 Intersecting Planes

Illustration of three intersecting planes. "The straight line perpendicular to one of two parallel lanes…

"If two straight lines are cut by three parallel planes, the corresponding segments are proportional."

3 Parallel Planes

"If two straight lines are cut by three parallel planes, the corresponding segments are proportional."

"If two angles not in the same plane have their sides respectively parallel and lying in the same direction, they are equal and their planes are parallel."

Angles In Parallel Planes

"If two angles not in the same plane have their sides respectively parallel and lying in the same direction,…