ON BODY
Translation © George MacDonald Ross, 1975–1999
PART I: CALCULATION OR LOGIC
Chapter 3: The Proposition
11. Categorical and hypothetical propositions.
12. The same proposition is expressed in many ways.
13. Propositions which can be reduced to the same categorical propositions are equipollent.
14. Universal propositions converted through contradictory names are equipollent.
15. Negative propositions are the same whether the negation comes before or after the copula.
16. Particular propositions simply converted are equipollent.
17. Subaltern, contrary, subcontrary, and contradictory propositions.
18. What consequence is.
20. How a proposition can be the cause of a proposition.
1. Definition of a syllogism.
2. There are only three terms in a syllogism.
3. The major, middle, and minor term, and the major and minor proposition.
4. In every syllogism, the middle term must be limited to one and the same thing in both propositions.
5. Nothing can be inferred from two particular propositions.
6. A syllogism is the addition of two propositions into a single sum.
7. What a syllogistic figure is.
9. How the first indirect figure is generated.
10. How the second indirect figure is generated.
11. How the third indirect figure is generated.
12. There are many modes in each figure, but mostly useless for philosophy.
13. A hypothetical syllogism is equivalent to a categorical one.
Chapter 13: Analogism, or the Same Ratio
1–4. The nature and definition of arithmetical and geometrical ratio.
5. The definition and various properties of the same arithmetical ratio.
6–7. The definition and transmutations of analogism, or the same geometrical ratio.
8–9. The definition and transmutations of hyperlogism and hypologism, that is, of larger or smaller ratio.
10–12. Comparisons of analogies of quantities in relation to magnitude.
13–15. Additions of ratios.
16–25. The definition and properties of continuous ratio.
26–29. The comparison between arithmetical and geometrical
ratio.
Chapter 14: Straight Lines, Curves, Angles, and Figures
1. The definition and properties of a straight line.
2. The definition and properties of a plane surface.
3. The species of curves.
4. The definition and properties of a circular line.
5. The properties of a straight line taken as in a plane.
6. The definition of contiguous lines.
7. The definition and species of an angle.
8. The angles of the same arc in concentric circles are proportional to the circumferences of the circles.
9. What the quantity of an angle consists in.
10. Different types of angles, taken in themselves.
11. Straight lines from the centre to the tangent of a circle.
12. The general definition of parallels, and the properties of straight parallels.
13. The ratio between the circumferences of circles is the same as the ratio between their diameters.
14. When two lines forming an angle are cut by two parallel straight lines, the ratio between the bases of the two triangles thus formed is the same as the ratio between their distances from the vertex along their sides.
15. What fraction of a straight line makes the circumference of a circle.
16. The angle between the circumference and a tangent is a quantity, but it is incommensurable with an angle in the strict sense, and cannot add anything to it or subtract anything from it.
17. The angle of inclination of a plane is an angle in the strict sense.
18. What a solid angle is.
19. The nature of asymptotes.
20. What things determine a position.
21. What a similar position is.
22. What a figure is, and what similar figures are.
PART III: THE RATIOS OF MOTIONS AND MAGNITUDES
Chapter 15: The Nature and Properties of Motion and Conation, and Various Other Considerations.
1. Summary of the principles of the theory of motion discussed earlier.
3. Certain theorems about the nature of motion.
4. Various considerations about motion.
5. The direction in which the first conation of moving bodies tends.
6. In motion arising from impact, if one ceases to move, the conation will be in the direction of the other.
8. The greater the speed or magnitude of a
moving body, the more effect its has on a body it hits.
Chapter 16: Accelerated and Uniform Motion, and Motion by Collision
1. The speed of a moving body, over whatever period of time it is calculated, is its impetus multiplied by the time.
2–5. In every motion, the distance covered is proportional to impetus multiplied by time.
6. If two bodies in uniform motion cover two distances, the distances covered will be in direct compound ratio of the ratios of time to time and impetus to impetus.
7. If two bodies in uniform motion cover two distances, the times taken will be in inverse compound ratio of the ratios of distance to distance and impetus to impetus. Similarly, the impetus of each will be in inverse compound ratio of the ratios of length to length and time to time.
8. If a body is moved simultaneously by two bodies in uniform motion, whatever angle they meet at, the line along which the body moves will be the straight line subtending the angle of the complement to two right angles.
9. If a body is moved simultaneously by two moving bodies,
the motion of one of which is uniform, and of the other of which is accelerated,
how to find its direction of motion, provided it is possible to quantify the
ratio between the distances covered and the times in which they are covered.
Chapter 17: Deficient Figures
1. The definitions of a deficient figure, of a complete figure, of the complement of a deficient figure, and of proportional and commensurable ratios.
2. The ratio of a deficient figure to its complement.
3. The ratios of deficient figures to the parallelograms in which they are inscribed presented in tabular form.
4. How these figures are drawn and extended.
5. How tangents to them are drawn.
6. The ratio by which these figures are larger than a rectilinear triangle with the same height and base.
7. A table of solid deficient figures inscribed in a cylinder.
8. The ratio by which these figures are larger than a cone with the same height and base.
9. How a plane deficient figure is inscribed within a parallelogram, in such a way that the ratio between it and a triangle with the same base and height, is the same as the ratio between twice another deficient figure (whether plane or solid) and the sum of the original deficient figure and the complete figure in which it is inscribed.
10. The application of certain properties of such figures drawn within parallelograms, to the ratios of distances covered at different speeds.
11. Deficient figures inscribed within circles.
12. The proposition demonstrated in Article 2 confirmed by first philosophy.
13. The equality between a section of the surface of a sphere and a circle.
14. How, by drawing deficient figures in a parallelogram,
any number of proportional middle lines can be interposed between two straight
lines.
Chapter 18: The Equation between Straight Lines and Paraboliform Lines
1. Given parabolic lines, to construct an equal straight line.
2. Given curved lines of the first parabolaster, or cubiform parabola, to find an equal straight line.
3. The general method for finding straight
lines equal to other curved lines belonging to the genus of parabola. [n.98]
Chapter 19: Equal Angles of Incidence and of Reflection
1. If two straight lines falling on a straight line are parallel, then the reflected lines will also be parallel.
2. If two straight lines coming from the same point fall on a straight line, their reflected lines extended in the other direction will meet at an angle which is equal to the angle between the original lines.
3. If two parallel straight lines fall on the circumference of a circle, their reflected lines extended inwards will make an angle which is twice that made by straight lines drawn from the centre to the points of incidence.
4. If two straight lines coming from the same point fall on the circumference of a circle, their reflected lines extended inwards will make an angle equal to twice the angle which is made by straight lines from the centre to the points of incidence, and the original lines.
5. If two straight lines coming from a point fall on the concave circumference, and make an angle which is less than twice the angle to the centre, their reflected lines will make an angle which, added to the angle of the original lines, will be equal to twice the angle to the centre.
6. If through any one point two unequal chords are drawn intersecting with each other, and the centre of the circle does not fall between them, and their reflected lines meet at some point, there cannot be drawn another straight line from point of origin of the first lines, the reflected line of which passes through the point of intersection between the first two reflected lines.
7. The same is not true if the chords are equal.
8. Given two points on the circumference of a circle, to draw two straight lines to those points in such a way that their reflected lines contain a given angle.
9. If a straight line falling on the circumference of a circle is extended to a radius, and if the part between the circumference and the radius is equal to the part of the radius between the point where they meet and the centre, the reflected line will be parallel to the radius.
10. If from a point within a circle two straight lines are drawn so that they fall on the circumference of the circle, and their reflected lines meet on the circumference of the circle itself, the angle between the reflected lines will be one third of the angle between the original lines.
1. In simple [circular] motion, any straight line selected in the moving body [n.107] is carried round in such a way that it is always parallel to its traces.
2. If there is circular motion around a fixed centre, and there is an epicycle on the circle, which revolves in the opposite direction in such a way that equal angles are generated in equal times, any straight line selected in the epicycle is carried round in such a way that it is always parallel to its traces.
3. The properties of simple [circular] motion.
4. If a liquid moves in simple circular motion, any points selected on it trace out their circles in times which are proportional to their distance from the centre.
5. Simple [circular] motion separates heterogeneous bodies, and collects homogeneous ones together.
6. If a circle traced out by something moving in simple motion is commensurable with a circle traced out by a point carried round with it, all the points on both circles will eventually return to the same position.
7. If a sphere is in simple [circular] motion, the extent to which it separates heterogeneous bodies is proportional to its distance from its poles.
8. If a solid body obstructs the simple circular motion of a liquid, there occurs an expansive motion of the liquid over the surface of the body.
9. Circular motion around a fixed centre throws off at a tangent any bodies which are on its circumference without being stuck to it.
10. Things which move with a simple circular motion generate simple circular motion.
11. If something which is moving this way is partly solid
and partly liquid, its motion will not be perfectly circular.
Chapter 22: Other Kinds of Motion
1. The difference between conation and effort.
2. Two kinds of media in which motion occurs.
3. The propagation of motion from one body to another.
4. The motion resulting from bodies pressing against each other.
5. Fluids pressed together penetrate each other.
6. A body which impinges upon another body, but without penetrating it, acts along a line which is drawn perpendicular to the surface of the body it impinges upon.
7. If a hard body penetrates another hard body which it impinges upon, it will penetrate it perpendicularly only if it hits it perpendicularly.
8. Sometimes a body can move in a direction opposite to that of the body moving it.
9. In a medium without any vacuum, motion is propagated to any distance.
10. Dilation and contraction.
11. Dilation and contraction presuppose that the minute parts change place.
13. Anything which reverts to its previous shape after being compressed or stretched has a motion of its internal parts.
14. If a vehicle stops, its load will not.
15–16. The effects of impact and of gravity seem to be incommensurable.
17–18. Motion cannot be initiated by the internal parts of a body.
19. Action and reaction occur along the same line.
Chapter 23: The Centre of Equilibrium of Things which Press Downwards in Parallel Straight Lines
1. Definitions and hypotheses.
2. No two planes of equilibrium are parallel.
3. There is a centre of equilibrium in every plane of equilibrium.
4. The downward force of equal weights is proportional to their distance from the centre of the weighing scales.
5–6. The downward force of unequal weights is in a compound ratio which is proportional to the weights and inversely proportional to their distance from the centre of the weighing scales.
7. If there are two weights, they are in equilibrium if their downward forces and distances from the centre of the weighing scales are inversely proportional, and vice versa.
8. If the parts of a weight all press equally on an arm of the scales, and individual parts are cut at a point measured from the centre of the scales, their downward forces will be in the same ratio as the parts of a triangle cut from the vertex by straight lines parallel to the base.
9. The diameter of equilibrium of figures which are deficient in respect of commensurable ratios of their heights and bases, divides the axis in such a way that the part at the vertex is to the rest, as the complete figure is to the deficient figure.
10. The diameter of equilibrium of the complement of half of any of the said deficient figures divides the line which is drawn through the vertex parallel to the base, in such a way that the part at the vertex is to the remainder, as the complete figure is to its complement.
11. The centre of equilibrium of half of any of the figures in the first row of the table in Chapter 17, Article 3, can be derived from the numbers in the second row.
12. The centre of equilibrium of half of any figure in the second row of the same table, can be derived from the numbers in the fourth row.
13. Once the centre of equilibrium of half of any figure in the same table is known, the centre of the area by which it is larger than a triangle with the same base and height is also known.
14. The centre of equilibrium of a solid sector is on its
axis, divided in such a way that the part at the vertex is to the whole axis,
minus half the axis of the portion of the sphere, is in a ratio of 3 to 4.
Chapter 24: Refraction and Reflection
1. Definitions.
2. There is no refraction in perpendicular motion.
3. Things projected from a rare medium into a dense one are refracted in such a way that the angle of refraction is greater than the angle of incidence.
4. A conation from a single point in every direction is refracted in such a way that the ratio of the sine of the angle of refraction to the sine of the angle of incidence is the inverse of the ratio of the density of the first medium to the density of the second medium.
5. The sine of the angle of refraction in one incidence is to the sine of the angle of refraction in another incidence, as the sine of the first angle of inclination is to the sine of the second angle of inclination.
6. If two lines of incidence have the same angle of incidence, and one is in a rare medium and the other in a dense medium, the sine of the angle of incidence will be the proportional mean of the sines of the two angles of refraction.
7. If the angle of incidence is half a right angle, and the line of incidence is in the denser medium, and the ratio between the densities is the same as that between the diagonal of a square and its side, and if the separating surface is plane, then the line of refraction will be on the separating surface itself.
8. If a body hits another body in a straight line, and does not penetrate it but is deflected, the angle of deflection will be equal to the angle of incidence.
9. The same happens in the generation of motion by impulse.
PART IV: PHYSICS, OR THE PHENOMENA OF NATURE
Chapter 26: The Universe and the Heavens
6. Possible causes of annual and daily rotation, and of the apparent forward motion, stopping, and backward motion of the planets.
7. A probable hypothesis as to the source of their simple motion.
8. The cause of the eccentricity of the annual motion of the earth.
9. The reason why the same side of the moon always faces the earth.
10. The cause of tides in the ocean.
11. The cause of the precession of the equinoxes.
Chapter 27: Light, Heat, and Colours
4. The generation of fire by the sun.
5. The generation of fire by collision.
6. The cause of light in fireflies, in rotten wood, and in phosphorus. [n.138]
7. The cause of light when sea water is struck.
8. The cause of flames, sparks, and melting.
9. The reason why wet hay sometimes combusts spontaneously, and the cause of lightning.
10. The cause of the power of gunpowder; and what is to be attributed to the charcoal, the sulphur, and the saltpetre.
11. How heat is generated by friction.
12. The distinction between primary light, secondary light, and so on.
13. The reason why the colours red, yellow, blue, and purple are seen through a prism.
14. Why the moon and the stars appear redder and larger at the horizon than in the middle of the sky.
15. The cause of whiteness.
16. The cause of blackness.
Chapter 28: Coldness, Wind, Hardness, Ice, Elasticity, Transparency, Thunder and Lightning, and the Origin of Rivers
1. Why breath from the same mouth sometimes warms and sometimes cools.
2. The origin of winds, and of their variability.
3. Why the wind from the East to the West near the Equator is gentle and unvarying.
4. The effect of air trapped in clouds.
5. The soft can become hard only because of motion.
6. The cause of coldness at the poles.
7. The cause of ice; why the cold is less severe when it rains than when the sky is clear; why water does not freeze in deep wells as readily as near the surface of the earth; why ice is lighter than water; and why wine freezes less readily than water.
8. Another cause of hardness is the more perfect contact of atoms; also how hard things are broken.
9. A third cause of hardness is the extraction of fluid parts.
10. A fourth cause of hardness is the motion of atoms in a confined space.
11. Another cause of hardness is heat; and how hard things become soft.
12. The source of the spontaneous restitution of elastic bodies.
13. The transparent and the opaque, and their source.
14. The cause of thunder and lightning.
15. The reason why clouds rise, freeze, and come down again.
16. How the moon could have been eclipsed when it was not seen in a position diametrically opposite the sun.
17. How a number of suns could be seen simultaneously.
18. The origin of rivers.
Chapter 29: Sound, Smell, Taste, and Touch
1. The definition of sound, and differences between sounds.
2. The cause of differences in degree among sounds.
3. The difference between low and high sounds.
4. The difference between clear and raucous sounds, and its source.
5. The source of the sound of thunder and of a cannon.
6. Why the flute has a clear sound when it is played.
7. Reflected sound.
8. The source of constant and long-lasting sound.
9. How sound can be increased or decreased by the wind.
10. Sound is transmitted not only by the air, but by any body, however hard.
11. The cause of low notes, high notes, and harmony.
12. Phenomena relating to smell.
13. The primary organ of smell, and the generation of smelling.
14. How it is reinforced by heat and wind.
15. Why bodies are less smelly if there is less ether intermingled with their parts.
16. Why scented things give off more scent when rubbed.
17. The organ of taste. And why some tastes are nauseating.
Chapter 30: Gravity
1. A dense body does not contain more matter in the same volume than a rarefied one.
2. Heavy bodies do not descend because of their own appetite, but because of a force exerted by the earth.
3. Differences in weight arise from the different forces with which the elements of heavy bodies press down towards the earth.
4. The cause of the descent of heavy bodies.
5. By what proportion the descent of heavy bodies is accelerated.
6. Why divers do not feel the weight of the water they are submerged in.
7. A body floating in water has the same weight as the quantity of water displaced by the immersed part.
8. If a body is lighter than water, then however big it is, it will float in any quantity of water, however small.
9. How it happens that water which has been sucked up into a vessel is forced out by air.
10. Why an inflated balloon is heavier than a deflated one.
11. The cause of the shooting upwards of heavy bodies from an air gun.
12. The cause of the rising of water in a thermometer.
13. The cause of the motion of living beings into the air.
14. There is in nature a type of body which is heavier than air, but not distinguishable from air by the senses.