An illustration depicting an astigmatism. An optical system with astigmatism is one where rays that propagate in two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances.

Diagram Illustrating Astigmatism

An illustration depicting an astigmatism. An optical system with astigmatism is one where rays that…

Diagram showing how to construct a conic when given the focus and the auxiliary circle. If the focus is outside the circle, we get a hyperbola. If it's inside the circle, we get an ellipse. If the auxiliary circle is a straight line (radius is infinite), we get a parabola.

Construction of a Conic

Diagram showing how to construct a conic when given the focus and the auxiliary circle. If the focus…

Diagram showing how to construct a conic when given the focus and the auxiliary circle. The focus is on the left of the auxiliary circle, thus producing a very obtuse hyperbola.

Focus In Auxiliary Circle of Conic

Diagram showing how to construct a conic when given the focus and the auxiliary circle. The focus is…

Diagram showing how to construct a conic when given the focus and the auxiliary circle. As the focus moves inside the circle the ellipse broadens out until the focus reaches the center and becomes a circle.

Focus In Auxiliary Circle of Conic

Diagram showing how to construct a conic when given the focus and the auxiliary circle. As the focus…

Illustration of half of an ellipse. "The ordinates of two corresponding points in an ellipse and its auxiliary circle are in the ratio b:a."

Corresponding Points in an Ellipse and Circle

Illustration of half of an ellipse. "The ordinates of two corresponding points in an ellipse and its…

Illustration showing that if one diameter is conjugate to a second, the second is conjugate to the first.

Conjugate Diameters of an Ellipse

Illustration showing that if one diameter is conjugate to a second, the second is conjugate to the first.

Illustration of half of an ellipse and its auxiliary circle used to construct an ellipse by points, having given its two axes.

Construction of an Ellipse

Illustration of half of an ellipse and its auxiliary circle used to construct an ellipse by points,…

Diagram showing how to construct an ellipse when given the two foci and the length of the major axis (2a).

Construction of an Ellipse

Diagram showing how to construct an ellipse when given the two foci and the length of the major axis…

Diagram of an ellipse that can used to illustrate the definition. "The constant ration between the distances of a point on an ellipse from the focus and the directrix equals the linear eccentricity divided by the semi major axis."

Definition of Ellipse

Diagram of an ellipse that can used to illustrate the definition. "The constant ration between the distances…

Illustration showing the definition of an ellipse. "An ellipse is a curve which is the locus of a point that moves in a plane so that the sum of its distances from two fixed points in the plane is constant." The foci and major axis is drawn.

Demonstration of Ellipse Definition

Illustration showing the definition of an ellipse. "An ellipse is a curve which is the locus of a point…

Illustration of half of an ellipse. "If d denotes the abscissa of a point of an ellipse, r and r' its focal radii, then r'=a+ed, r=a-ed."

Focal Radii of an Ellipse

Illustration of half of an ellipse. "If d denotes the abscissa of a point of an ellipse, r and r' its…

Diagram that illustrates "The algebraic sum of the distances of any point from the foci of a conic is greater or less than 2a, according as the point is without or within the curve."

Foci Distances on an Ellipse

Diagram that illustrates "The algebraic sum of the distances of any point from the foci of a conic is…

A graph of tan ellipse with Foci and Center.

Graph of Ellipse with Foci and Center Labeled

A graph of tan ellipse with Foci and Center.

Illustration of an ellipse with foci F' and F, major axis A' to A, minor axis B' to B, and center O.

Ellipse With Parts Labeled

Illustration of an ellipse with foci F' and F, major axis A' to A, minor axis B' to B, and center O.

Illustration of half of an ellipse. "If through a point P of an ellipse a line is drawn bisecting the angle between one of the focal radii and the other produced, every point in this line except P is without the curve."

Line Bisecting Angle Between Focal Radii on Ellipse

Illustration of half of an ellipse. "If through a point P of an ellipse a line is drawn bisecting the…

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…

Illustration showing that tangents drawn at the ends of any diameter are parallel to each other.

Parallel Tangents to an Ellipse

Illustration showing that tangents drawn at the ends of any diameter are parallel to each other.

Illustration of ellipse with parts labeled.

Ellipse With Parts Labeled

Illustration of ellipse with parts labeled.

Diagram of an ellipse that can used to illustrate the different parts. Segment MN is the major axis, segment CD is the conjugate (minor) axis, and point O is the center of the ellipse. Both foci are also labeled in the illustration.

Parts of Ellipse

Diagram of an ellipse that can used to illustrate the different parts. Segment MN is the major axis,…

Illustration of half of an ellipse. "The sum of the distances of any point from the foci of an ellipse is greater than or less than 2a, according as the point is without or within the curve."

Point Distances to Foci on Ellipse

Illustration of half of an ellipse. "The sum of the distances of any point from the foci of an ellipse…

Illustration of how to draw a tangent to an ellipse from an external point.

Tangent From External Point to an Ellipse

Illustration of how to draw a tangent to an ellipse from an external point.

Diagram an ellipse with a tangent line that illustrates "A line through a point on the ellipse and bisecting the external angle between the focal radii is a tangent."

Tangent to an Ellipse

Diagram an ellipse with a tangent line that illustrates "A line through a point on the ellipse and bisecting…

Illustration showing the tangents drawn at two corresponding points of an ellipse and its auxiliary circle cut the major axis produced at the same point.

Tangents to an Ellipse

Illustration showing the tangents drawn at two corresponding points of an ellipse and its auxiliary…

A section of the human eye.

Eye

A section of the human eye.

"Diagram showing the Change in the Lens during Accomadation. On the right the lens is arranged for distant vision, the cilliary muscle is relaxed, and the ligament <em>D</em> is tense, so flattening by its compression the front of the lens <em>C</em>, on the left the muscle <em>A</em> is acting, and this relaxes the ligament and allows the lens <em>B</em> to become more convex, and so fitted for the vision of near objects." &mdash; Blaisedell, 1904

Lens of the eye

"Diagram showing the Change in the Lens during Accomadation. On the right the lens is arranged for distant…

"A fire-place; a hearth; a brazier. The fire-place possessed a sacred character, and was dedicated among the Romans to the Lares of each family. Movable hearths, or braziers, properly called foculi, were frequently used." &mdash; Smith, 1873.

Focus

"A fire-place; a hearth; a brazier. The fire-place possessed a sacred character, and was dedicated among…

An illustration depicting the formation of circles of diffusion. "From point A luminous rays enter the eye in the form of a cone, the kind of which will depend on the pupil. Thus it may be circular, or oval, or ever triangular. If the pencil is focused in front of the retina, as at d, or behind it as it as at f, or, in other words, if the retina of being at F; be in the position G or H, there will be a luminous circle or a luminous triangular space, and many elements of the retina will be affected. The size of these diffusion circles depends on the distance from the retina of the point where the rays are focused: the greater the distance, the more extended will be the diffusion circle" (Britannica, 132).

Formation of Circles of Diffusion

An illustration depicting the formation of circles of diffusion. "From point A luminous rays enter the…

"Heat is radiated at C, which is placed at mirror A's focus. Mirror A catches some of the heat and reflects it across to mirror B which focuses the heat on D." &mdash;Quackenbos 1859

Reflection of Radiant Heat

"Heat is radiated at C, which is placed at mirror A's focus. Mirror A catches some of the heat and reflects…

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…

Diagram depicting a cone with both nappes intersected by plane J to form a hyperbola.

Cone Intersected by a Plane to Form a Hyperbola

Diagram depicting a cone with both nappes intersected by plane J to form a hyperbola.

Illustration showing how the a hyperbola is symmetrical with respect to its conjugate axis.

Conjugate Axis of a Hyperbola

Illustration showing how the a hyperbola is symmetrical with respect to its conjugate axis.

Illustration showing how a hyperbola con be constructed by points, having been given the foci and the constant difference 2a.

Construction of Hyperbola

Illustration showing how a hyperbola con be constructed by points, having been given the foci and the…

Diagram showing how to construct a hyperbola when given the two foci and the length of the major axis (2a).

Construction of a Hyperbola

Diagram showing how to construct a hyperbola when given the two foci and the length of the major axis…

Illustration showing the definition of an hyperbola. "An hyperbola may be described by the continuous motion of a point, as follows: To one of the foci F' fasten one end of a rigid bar F'B so that it is capable of turning freely about F' as a center in the plane of the paper."

Demonstration of Hyperbola Definition

Illustration showing the definition of an hyperbola. "An hyperbola may be described by the continuous…

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…

Diagram part of a hyperbola with a tangent line that illustrates "A line through a point on the hyperbola and bisecting the internal angle between the focal radii is a tangent."

Tangent to a Hyperbola

Diagram part of a hyperbola with a tangent line that illustrates "A line through a point on the hyperbola…

"A concave mirror reflects from a curved surface hollowing in like he inside of an orange." &mdash;Quackenbos 1859

Concave Mirror

"A concave mirror reflects from a curved surface hollowing in like he inside of an orange." —Quackenbos…

"The focus of a concave mirror is the point where the rays are brought together by reflection." -Comstock 1850

Focus of a Concave Mirror

"The focus of a concave mirror is the point where the rays are brought together by reflection." -Comstock…

"Parallel rays of light strike the concave mirror. The rays converge at the focus, F, which is halfway between the mirror surface and the center of the sphere that the mirror would form if it were a full sphere." &mdash;Quackenbos 1859

Reflection from Concave Mirrors

"Parallel rays of light strike the concave mirror. The rays converge at the focus, F, which is halfway…

"...be made obvious...where the diverging rays 1, 2, 3, 4 form a focus at the point o, whereas, had they been parallel, their focus would have been at a." -Comstock 1850

Divergent Rays in a Concave Mirror

"...be made obvious...where the diverging rays 1, 2, 3, 4 form a focus at the point o, whereas, had…

"When the rays diverge from a point beyond the center of curvature, as B, the focus falls on the same axis, at a distrance from the mirror greater than that of the principal focus, and less than that of the center of curvature." -Avery 1895

Rays Diverging from Beyond the Center of Curvature on a Concave Mirror

"When the rays diverge from a point beyond the center of curvature, as B, the focus falls on the same…

"When the rays diverge form a point at a distance from the mirror less than that of the principal focus, the reflected rays diverge as if from a point back of the mirrir. This point, b, is a virtual focus." -Avery 1895

Rays Diverging from Beyond the Center of Curvature on a Concave Mirror

"When the rays diverge form a point at a distance from the mirror less than that of the principal focus,…

"The focus of each point chose may be determined by tracing two rays from the point, and locating their real or apparent intersection after reflection by the mirror. The two rays most convenient for this purpose are teh oen that lies along the axis fo the point, and the one that lies parallel to the principal axis of the mirror." -Avery 1895

Concave Mirror with Image and Focus

"The focus of each point chose may be determined by tracing two rays from the point, and locating their…

"When the object is at a distance from the mirror somewhat greater than the center of curvature, as beyond C, the image is real, inverted, smaller than the object, and at a distance from the mirror greater than that of the principle focus and less than that of the center of curvature, as between F and C." -Avery 1895

Image Beyond the Curvature of a Concave Mirror

"When the object is at a distance from the mirror somewhat greater than the center of curvature, as…

"...if the object is placed more remote from the mirror than the principal focus, and between the focus and the centre of the sphere of which the reflector is a part, then the image will appear inverted on the contrary side of the centre, and farter from the mirror than the object; thus, if a lamp be placed obliquely before a concave mirror, its image will be seem inverted in the air, on the contrary side of a perpendicular line through the centre of the mirror." -Comstock 1850

Object Beyond the Focus in a Concave Mirror

"...if the object is placed more remote from the mirror than the principal focus, and between the focus…

"...let us suppose the object a, to be placed before the mirror, and nearer to it than the principal focus. Then the rays proceeding from the extremities of the object without interruption, would continue to diverge in the lines o and n, as seen behind the mirror' but by reflection they are made to diverge less than before, and consequently to make the angle under which the meet more obtuse at the eye b, than it would be if they continued onward to e, where they would have met without reflection. The result therefore, is to render the image h, upon the eye, as much larger than the object a, as the angle at the eye is more obtuse than the angle at e." -Comstock 1850

Object Within the Focus in a Concave Mirror

"...let us suppose the object a, to be placed before the mirror, and nearer to it than the principal…

"Suppose the tumbler, a, to be filled with water, and placed beyond the principal focus of the concave mirror, and so managed as to be hid from eye c, by the screen b. The lamp by which the tumbler is illuminated must also be placed behind the screen, and near the tumbler, To a person placed at c, the tumbler with its contents will appear incerted at e, and suspended in the air." -Comstock 1850

Deception by Mirrors

"Suppose the tumbler, a, to be filled with water, and placed beyond the principal focus of the concave…

"A convex mirror reflects from a curved surface rounding out like the outside of an orange." &mdash;Quackenbos 1859

Convex Mirror

"A convex mirror reflects from a curved surface rounding out like the outside of an orange." —Quackenbos…

"Parallel rays strike the convex mirror, reflect, and diverge as if they had originated from a virtual focus inside the mirror. Focus F is located between the surface of the mirror and the mirror's center if it were a full body sphere." &mdash;Quackenbos 1859

Reflection by Convex Mirrors

"Parallel rays strike the convex mirror, reflect, and diverge as if they had originated from a virtual…

Two concave mirrors facing each other to concentrate light.

Conjugate Mirrors

Two concave mirrors facing each other to concentrate light.

The segments between the point of intersection of two tangents to a conic and their points of contact are seen from a focus under equal angles. the ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant.

Parabola Foci Properties

The segments between the point of intersection of two tangents to a conic and their points of contact…

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…

Illustration showing the definition of a parabola as a conic section. "The section of a right circular cone made by a plane parallel to one, and only one, element of the surface is a parabola."

Conic Section Showing Parabola

Illustration showing the definition of a parabola as a conic section. "The section of a right circular…

Illustration showing the definition of a parabola as a conic section. "The section of a right circular cone made by a plane parallel to one, and only one, element of the surface is a parabola."

Conic Section Showing Parabola

Illustration showing the definition of a parabola as a conic section. "The section of a right circular…

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…

Illustration of a parabola showing that any point of a parabola is the mean proportional between the latus rectum (focal chord) and the abscissa (x-coordinate).

Parabola With Coordinates and Latus Rectum

Illustration of a parabola showing that any point of a parabola is the mean proportional between the…

Illustration of a parabola - a curve which is the locus of a point that moves in a plane so that its distance from a fixed point in the plane is always equal to its distance from a fixed line in the plane.

Parabola With Focus and Directrix

Illustration of a parabola - a curve which is the locus of a point that moves in a plane so that its…