Figure 1
Drawing of the Statue of Liberty by a child of three years old.
Figure 2
A prehistoric rock painting of a family of capybaras in the Serra da Capivara National Part.
Figure 3
At the left, a painting by Magritte warning: this is not a pipe. Indeed, it is a representation of a pipe. At the right, the Maxwell equations, which are not light but a representation of light.
Figure 4
Illustration of the structure of cognition. The thinking abilities are innate structures of the mind. The symbolic framework is created and developed by the thinking abilities and is a representation of the real world.
Figure 5
Illustration of the structure of the scientific knowledge. Rational thinking is an innate structure of the mind. The scientific theory is constructed by rational thinking and is a representation of the real phenomena.
Figure 6
Illustration of Euclid’s fifth postulate. If the sum of the angles α and β is strictly less than two right angles, then the blue straight lines will meet, if extended indefinitely.
Figure 7
Illustration of proposition 47 of book 1 of the Elements, which is the Pythagorean theorem. The area of the square BCED is equal to the sum of the areas of the squares ABFG and ACKH.
Figure 8
When the moon appears exactly in half phase, the angle earth-moon-sun is 90°. The value used by Aristarchus for the angles β is 87° which gives α equal to 3°. The modern value of α is 10′.
Figure 9
The diagram shows the arrangement of the sun, the earth, and the moon at a lunar eclipse. The points S, E, and M, which are the centers of the sun, the earth, and the moon, are collinear. The straight line ABC is tangent to the circumferences, and DB and FC are parallel to SEM.
Figure 10
Illustration of the demonstration of the Archimedes law of lever for n=2 and m=3. The weight of the big yellow block is equal to the three small yellow bricks. The weight of the big blue block is equal to the two small blocks. The small bricks are identical and are equally spaced.
Figure 11
This mobile after Calder illustrates the Archimedes law of the lever. Each small circle is located at the vertical passing through the center of gravity of the mobile attached to the circle. If a mobile is replace by another one with the same weight, the equilibrium is unaltered.
Figure 12
A segment of a parabola AOB with vertex at O and axis OC. The inclined dashed line is parallel to AB and tangent to the parabola at F. The center of gravity G is on the parallel FH to the axis at F.
Figure 13
ACBO is a paraboloid of revolution around the axis OC, and G is its center of gravity. F is the center of gravity of the part of the paraboloid inside the fluid. In the left panel F and G are in the same vertical, which is the condition for equilibrium. In the right panel the paraboloid is inclined, and F is at the left of the vertical dashed line passing through G, an arrangement that restores the equilibrium.
Figure 14
The trajectory of the sun S is the eccentric circumference centered at point O. It moves at constant speed along the circumference which means that the angle SOA increases uniformly. The earth T is off the center. The angle STA, which is the angle an observer sees the sun with respect to A, varies non-uniformly.
Figure 15
The point G revolves counterclockwise around the earth T along the great circumference at a constant angular velocity. The moon L moves clockwise along the small circle around the point G at a constant angular velocity.
Figure 16
The planet P orbits counterclockwise along the epicycle centered at the point G which revolves along the deferent circumference centered at O. The earth T is off the center and E is the equant point.
Figure 17
Reflection on a plane and on a spherical mirror. The light ray coming from the object O is reflected at the point R of the reflecting surface and reaches the point S. The image I is formed at the point where the cathetus OI crosses the extension of the reflected ray.
Figure 18
Refraction through a plane and through a spherical interface. The light ray coming from the object O is refracted at the point R of the interface and reaches the point S. The image I of the object O is formed at the point where the cathetus OI crosses the extension of the refracted ray.
Figure 19
Illustration of the law of refraction as formulated by Ibn Sahl. OR and RS are the incident and the refracted rays and BR is the extension of RS. The points B and C are located at the vertical passing through O. The length of AR is equal to OR. The ratio RA/RB is equal to the ratio RC/RK.
Figure 20
The trajectory of the earth T is the eccentric circumference centered at point O. It moves at constant speed along the circumference which means that the angle TOA increases uniformly. The sun S is off the center and the angle TSA varies non-uniformly.
Figure 21
Earth T, Venus V, and Mars M revolving around the sun S. Venus has an inner orbit whereas Mars has an outer orbit with relation to the earth orbit. A, B, C are three distinct positions of the earth. The position A corresponds to the maximum value of the angle SAV. The position B occurs when Mars is in opposition to the sun, and position C occurs when the angle SCM is equal to a right angle.
Figure 22
The earth T revolves along the eccentric circumference around the sun S. The points A, B, and C represents the positions of the earth at three instants such that the mars M is at the same position.
Figure 23
The phases of Venus at three positions A, B, and C of the planet with respect to the sun S and the earth T. At the right the phases of Venus as seen from the earth at the three positions.
Figure 24
AB represents the time and BC the final velocity of a uniformly accelerated motion. (a) Representation used by Galilleo. (b) Representation used by Huygens.
Figure 25
The final velocity of a body starting from rest descending the three inclined planes or any descending curve depends only on the height.
Figure 26
At left, AB and BF are the incident and reflect rays at the point B of the reflecting surface CBE. The point B is the center of the circumference. At right, AB and BI are the incident and refracted rays at the point B of the interface CBE of two media. The point B is the center of the two circumferences.
Figure 27
An illustration contained in the Meteorology of Descartes [1515. R. Descartes, Discours de la Methode, la Dioptrique, les Meteores et la Geometrie (Maire, Leyde, 1637).] explaining how the rainbow is formed. A light ray coming from the sun suffers an internal reflection in a drop of water and reaches the observer’s eye. The principal arc of the rainbow is produced by the ray ABCDE, which undergoes a single internal reflection, whereas the secondary arc is produced by the ray FGHIKE, which undergoes two internal reflections.
Figure 28
A spherical drop of water. At left, the light ray suffers one internal reflections, giving rise to the principal arc of the rainbow. At right, the light ray suffers two internal reflection, giving rise to the secondary arc of the rainbow.
Figure 27
An illustration contained in the Meteorology of Descartes [1515. R. Descartes, Discours de la Methode, la Dioptrique, les Meteores et la Geometrie (Maire, Leyde, 1637).] explaining how the rainbow is formed. A light ray coming from the sun suffers an internal reflection in a drop of water and reaches the observer’s eye. The principal arc of the rainbow is produced by the ray ABCDE, which undergoes a single internal reflection, whereas the secondary arc is produced by the ray FGHIKE, which undergoes two internal reflections.
Figure 28
A spherical drop of water. At left, the light ray suffers one internal reflections, giving rise to the principal arc of the rainbow. At right, the light ray suffers two internal reflection, giving rise to the secondary arc of the rainbow.
Figure 29
Illustration of the Pascal hydrostatic principle. At left, four vessels with distinct shapes but with the same base. The pressure at the bottom is the same in all the four vessels. At right, the small body equilibrates the large body.
Figure 30
(a) Parabolic trajectory ABC of a projectile B launched horizontally from point A. The motion is decomposed into a uniform motion AD and a uniformly accelerated motion DB. (b) Circular trajectory of a body H tied to a thread HN over a horizontal plane. The motion is decomposed into a uniform motion GL and a uniformly accelerated motion LH.
Figure 31
(a) Simple pendulum: the trajectory ABC is an arc of circumference. (b) Cycloidal pendulum: the trajectory DEF is a cycloid and the period of oscillation is independent of the amplitude.
Figure 32
The curve OAF is a cycloid. A body released from rest at any point of the trajectory, such as the point G, will take the same time to reach the lowest point F. Any point A of the cycloid OAF is mapped onto a point P of the semicircumference OPM. Any point of the trajectory GAF is mapped onto the small semicircumference LBF.
Figure 33
Illustration of the Huygens principle. BC and DE are wave fronts originating from the point A, whereas the small arcs are secondary front waves originating from the points on the arc BC.
Figure 34
Reflection of light waves. AB is an incident front wave and CD is the reflection front wave. CE is an arc of the circumference with center at A and radius AC equal to BD and understood as a secondary wave front with origin at A.
Figure 35
Refraction of light waves. AB is an incident front wave and CD is the refraction front wave. CE is an arc of the circumference centered at A and radius equal to AC which is equal to BD times the ratio between the wave velocities of the lower and upper media. CE is understood as a secondary wave front with origin at A.
Figure 36
AB and BC are the incident and refracted rays at the point B of an interface BF. AFC is another arbitrary path connecting the points A and C. The time to reach point C from A is smaller for the path ABC, which obey the sine law, than it is for the path AFC.
Figure 31
(a) Simple pendulum: the trajectory ABC is an arc of circumference. (b) Cycloidal pendulum: the trajectory DEF is a cycloid and the period of oscillation is independent of the amplitude.
Figure 32
The curve OAF is a cycloid. A body released from rest at any point of the trajectory, such as the point G, will take the same time to reach the lowest point F. Any point A of the cycloid OAF is mapped onto a point P of the semicircumference OPM. Any point of the trajectory GAF is mapped onto the small semicircumference LBF.
Figure 33
Illustration of the Huygens principle. BC and DE are wave fronts originating from the point A, whereas the small arcs are secondary front waves originating from the points on the arc BC.
Figure 34
Reflection of light waves. AB is an incident front wave and CD is the reflection front wave. CE is an arc of the circumference with center at A and radius AC equal to BD and understood as a secondary wave front with origin at A.
Figure 35
Refraction of light waves. AB is an incident front wave and CD is the refraction front wave. CE is an arc of the circumference centered at A and radius equal to AC which is equal to BD times the ratio between the wave velocities of the lower and upper media. CE is understood as a secondary wave front with origin at A.
Figure 36
AB and BC are the incident and refracted rays at the point B of an interface BF. AFC is another arbitrary path connecting the points A and C. The time to reach point C from A is smaller for the path ABC, which obey the sine law, than it is for the path AFC.
Figure 37
When the point B approaches the point A along the curve ABC, the ultimate ratios of the lengths of the arc AB, the cord AB and the tangent AD are equal to unity. When the widths of the rectangles decrease and their number increase in infinitum, the sum of the areas of the rectangles approaches the area under the curve GH.
Figure 38
The trajectory of a body under the action of a central force toward the center S is approximated by a polygonal path. The points A, B, C, D, and E are chosen so that the intervals of time for traveling each line segment are the same. ABCK is a parallelogram and K is on the line SB.
Figure 39
A body P describes the ellipse ABGK and is under an attractive central force directed towards center C of the ellipse. The line RP is tangent to the ellipse at P and QV and the diameter DCK are parallel to RP, and PF is perpendicular to RP. The line QR is parallel to the line CP joining P to the center C of the ellipse and QT is perpendicular to CP.
Figure 40
A body P describes the ellipse ABGK and is under an attractive central force directed towards the focus S of the ellipse. The line RP is tangent to the ellipse at P and QV and the diameter DCK are parallel to RP. The line QR is parallel to the line CS joining P to the focus S of the ellipse and QT is perpendicular to CS.
Figure 41
AEFB is a cross section of a spherical surface with center at S. The mass is distributed uniformly on the surface. Each element of the surface attracts a corpuscle placed at the point P with a force inversely proportional to the square of the distance. The lines PE and PF join the ends of the element EF to the point P, and EC and FD are perpendicular to PS.
Figure 42
Orbit of the great comet of 1680 taken from an illustration of the Principles [8787. F. Cajori, Sir Isaac Newton’s Mathematical Principles (University of California Press, Berkeley, 1974), 2 v.]. ABC represents the orbit of the comet, D the sun, GH the intersection with the sphere of the orbit of the earth. The points I, K, L, M, N, O, P, Q, R, S, T, and V represent the place of the comet at November 4, 11, 19, December 12, 21, 29, 1680, January 5, 25, February 5, 25, March 5, 9, 1681, respectively.
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