Top 15 Colin MacLaurin Quotes

Nature … has made it impossible for us to have any communication from this earth with the other great bodies of the universe, in our present state; and it is highly possible that he has likewise cut off all communication betwixt the other planets, and betwixt the different systems.… We observe, in all of them, enough to raise our curiosity, but not to satisfy it … It does not appear to be suitable to the wisdom that shines throughout all nature, to suppose that we should see so far, and have our curiosity so much raised … only to be disappointed at the end … This, therefore, naturally leads us to consider our present state as only the dawn or beginning of our existence, and as a state of preparation or probation for farther advancement.…

He [Kepler] supposes, in that treatise [epitome of astronomy], that the motion of the sun on his axis is preserved by some inherent vital principle; that a certain virtue, or immaterial image of the sun, is diffused with his rays into the ambient spaces, and, revolving with the body of the sun on his axis, takes hold of the planets and carries them along with it in the same direction; as a load-stone turned round in the neighborhood of a magnetic needle makes it turn round at the same time. The planet, according to him, by its inertia endeavors to continue in its place, and the action of the sun's image and this inertia are in a perpetual struggle. He adds, that this action of the sun, like to his light, decreases as the distance increases; and therefore moves the same planet with greater celerity when nearer the sun, than at a greater distance. To account for the planet's approaching towards the sun as it descends from the aphelium to the perihelium, and receding from the sun while it ascends to the aphelium again, he supposes that the sun attracts one part of each planet, and repels the opposite part; and that the part which is attracted is turned towards the sun in the descent, and that the other part is towards the sun in the ascent. By suppositions of this kind he endeavored to account for all the other varieties of the celestial motions.

But to return to Kepler, his great sagacity, and continual meditation on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual betwixt bodies, and tells us that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds that the tides arise from the gravity of the waters towards the moon. But not having just enough notions of the laws of motion, he does not seem to have been able to make the best use of these thoughts; nor does he appear to have adhered to them steadily, since in his epitome of astronomy, published eleven years after, he proposes a physical account of the planetary motions, derived from different principles.

They proceeded therefore in another manner, less direct indeed, but perfectly evident. They found, that the inscribed similar polygons, by increasing the number of their sides, continually approached to the areas of the circles; so that the decreasing differences betwixt each circle and its inscribed polygon, by still further and further divisions of the circular arches which the sides of the polygons subtend, could become less than any quantity that can be assigned: and that all this while the similar polygons observed the same constant invariable proportion to each other, viz. that of the squares of the diameters of the circles. Upon this they founded a demonstration, that the proportion of the circles themselves could be no other than that same invariable ratio of the similar inscribed polygons; of which we shall give a brief abstract, that it may appear in what manner they were able... to form a demonstration of the proportions of curvilineal figures, from what they had already discovered of rectilineal ones. And that the general reasoning by which they demonstrated all their theorems of this kind may more easily appear, we shall represent the circles and polygons by right lines, in the same manner as all magnitudes are expressed in the fifth book of the Elements.

Circles are the only curvilineal plane figures considered in the elements of geometry. If they could have allowed... these as similar polygons of an infinite number of sides (as some have done who pretend to abridge their demonstrations), after proving that any similar polygons inscribed in circles are in the duplicate ratio of the diameters, they would have immediately extended this to the circles themselves and would have considered the second proposition of the twelfth book of the Elements as an easy corollary from the first. But there is ground to think that they would not have admitted a demonstration of this kind. It was a fundamental principle with them, that the difference of any two unequal quantities, by which the greater exceeds the lesser, may be added to itself till it shall exceed any proposed finite quantity of the same kind: and that they founded their propositions concerning curvilineal figures upon this principle... is evident from the demonstrations, and from the express declaration of Archimedes, who acknowledges it to be the foundation...[of] his own discoveries, and cites it as assumed by the antients in demonstrating all their propositions of this kind. But this principle seems to be inconsistent with... admitting... an infinitely little quantity or difference, which, added to itself any number of times, is never supposed to become equal to any finite quantity whatsoever.

They found, that similar triangles are to each other in the duplicate ratio of their homologous sides; and, by resolving similar polygons into similar triangles, the same proposition was extended to these polygons also. But when they came to compare curvilineal figures, that cannot be resolved into rectilineal parts, this method failed.

[C]onsider the steps by which the antients were able... from the mensuration of right-lined figures, to judge of such as were bounded by curve lines; for as they did not allow themselves to resolve curvilineal figures into rectilineal elements, it is worth examin[ing] by what art they could make a transition from the one to the other: and as they... finish their demonstrations in the most perfect manner... by following their example... in demonstrating a method so much more general than their's, we may best guard against exceptions and cavils, and vary less from the old foundations of geometry.

The method of demonstration, which was invented by the author of fluxions, is accurate and elegant; but we propose to begin with one that is somewhat different; which, being less removed from that of the antients, may make the transition to his method more easy to beginners (for whom chiefly this treatise is intended), and may obviate some objections that have been made to it.

In delivering the principles of this method, we apprehend it is better to avoid such suppositions: but after these are demonstrated, short and concise ways of speaking, though less accurate, may be permitted, when there is no hazard of our introducing any uncertainty or obscurity into the science from the use of them, or of involving it in disputes.

We shall not consider any part of space or time as indivisible, or infinitely little; but we shall consider a point as a term or limit of a line, and a moment as a term or limit of time: nor shall we resolve curve lines, or curvilineal spaces, into rectilineal elements of any kind.

When the certainty of any part of geometry is brought into question, the most effectual way to set the truth in a full light, and to prevent disputes, is to deduce it from s or first principles of unexceptionable evidence, by demonstrations of the strictest kind, after the manner of the antient geometricians. This is our design in the following treatise; wherein we do not propose to alter Sir Isaac Newton's notion of a , but to explain and demonstrate his method, by deducing it at length from a few self-evident truths, in that strict manner: and, in treating of it, to abstract from all principles and postulates that may require the imagining any other quantities but such as may be easily conceived to have a real existence.

He considered magnitudes as generated by a or motion, and showed how the velocities of the generating motions were to be compared together. There was nothing in this doctrine but what seemed to be natural and agreeable to the antient geometry. But what he has given us on this subject being very short, his conciseness maybe supposed to have given some occasion to the objections which have been raised against his method.

There were some, however, who disliked the... use of infinites and infinitesimals in geometry. Of this number was Sir Isaac Newton (whose caution was almost as distinguishing a part of his character as his invention), especially after he saw that this liberty was growing to so great a height. In demonstrating the grounds of the method of fluxion, he avoided them, establishing it in a way more agreeable to the strictness of geometry.

This way of considering what is called the sublime part of geometry has so far prevailed, that it is generally known by no less a title than the Science, Arithmetic, or Geometry of infinites. These terms imply something lofty, but mysterious; the contemplation of which may be suspected to amaze and perplex, rather than satisfy or enlighten the understanding... and while it seems greatly to elevate geometry, may possibly lessen its true and real excellency, which chiefly consists in its perspicuity and perfect evidence; for we may be apt to rest in an obscure and imperfect knowledge of so abstruse a doctrine... instead of seeking for that clear and full view we ought to have of geometrical truth; and to this we may ascribe the inclination... of late for introducing mysteries into a science wherein there ought to be none.

After these came to be relished, an infinite scale of infinites and s (ascending and descending always by infinite steps) was imagined and proposed to be received into geometry, as of the greatest use for penetrating into its abstruse parts. Some have argued for quantities more than infinite; and others for a kind of quantities that are said to be neither finite nor infinite, but of an intermediate and indeterminate nature.