Colin MacLaurin Quotes

A <small>LETTER</small> published in the year 1734, under the title of first gave occasion to the ensuing Treatise; and several reasons concurred to induce me to write on this subject at so great a length. The Author of that Piece had represented the as founded on false Reasoning, and full of Mysteries. His Objections seemed to have been occasioned in a great measure, by the concise manner in which the Elements of this Method have been usually described; and their having been so much misunderstood by a person of his abilities, appeared to me a sufficient proof that a fuller Account of the Grounds of them was requisite.

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.

I accommodated my Definition of the Variation of Curvature in Chap. xi. to Sir Isaac Newton's, to prevent mistakes, as I have observed in Article 386, but made no material alteration in any thing else.

In the method of indivisibles, lines were conceived to be made up of points, surfaces of lines,and solids of surfaces; and such suppositions have been employed by several ingenious men for proving the old theorems, and discovering new ones, in a brief and easy manner. But as this doctrine was inconsistent with the strict principles of geometry, so it soon appeared that there was some danger of its leading them into false conclusions: therefore others, in the place of indivisible, substituted infinitely small divisible elements, of which they supposed all magnitudes to be formed; and thus endeavoured to retain, and improve, the advantages that were derived from the former method for the advancement of geometry.

These, with other observations concerning this method, and its application, led me on gradually to compose a Treatise of a much greater extent than I intended, or would have engaged in, if I had been aware of it when I began this Work, because my attendance in the University could allow one to bestow but a small part of my time in carrying it on.

But it has been objected on several occasions, that the modern improvements have been established for the most part upon new and exceptionable maxims, of too abstruse a nature to deserve a place amongst the plain principles of the ancient geometry: and some have proceeded so far as to impute false reasoning to those authors who have contributed most to the late discoveries, and have at the same time been most cautious in their manner of describing them.

This determined me, immediately after that Piece came to my hands, and before I knew any thing of what was intended by others in answer to it, to attempt to deduce those Elements after the manner of the Antients, from a few unexceptionable principles, by Demonstrations of the strictest form.

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.…

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.

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 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.

I perceived that some Rules were defective or inaccurate; that the Resolution of several Problems which had been deduced in a mysterious manner, by second and third s, could be completed with greater evidence, and less danger of error, by first Fluxions only; and that other problems had been resolved by Approximations, when an accurate Solution could be obtained with the same or greater facility.

In explaining the Notion of & , I have followed Sir Isaac Newton in the first Book, imagining that there can be no difficulty in conceiving Velocity wherever there is Motion; nor do I think that I have departed from his Sense in the second Book; and in both I have endeavoured to avoid several expressions, which, though convenient, might be liable to exceptions, and, perhaps, occasion disputes. I have always represented Fluxions of all... Orders by finite Quantities, the Supposition of an infinitely little Magnitude being too bold a Postulatum for such a Science as Geometry.

Several Treatises have appeared while this was in the press, wherein some of the same Problems have been considered, though generally in a different manner. I have had occasion to mention most of them in the last Chapter of the second Book; but had not there an opportunity to take notice, that the Problem in 480 has been considered by Mr. Euler in his Mechanics.

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.