DISSERTATION Concerning the nature and extent of geometrical IN 7 N the former differtation, as much care as poffible has been taken to avoid the suggesting any circumftance to the reader's imagination, which will not be neceffary in the very first steps to be taken in this science. I must therefore beg he will not confider what has been faid as words of courfe, merely to scrape acquaintance, or any attempt to display my own learning, but as seriously. intended for his improvement. Nor would I have it thought that this proceeds from any want of inclination to be in the good graces of my reader, much less from want of capacity; for with very confiderable deductions both of thought and trouble, like fome other commentators I could have flourished away, in an explanation of the eleventh common notion, upon the properties of the afymptotes of the Hyperbola &c; if my intention had been to raise the admiration of the reader, rather than to fix his attention. Having therefore refolved to facrifice every ornament to this fingle confideration ; the learner, it is to be hoped, will endeavour to convince himself, or rather fome more impartial judge, that he understands every thing mentioned in the last differtation before he proceeds to this; in which he will find the nature and end of geometrical demonftration laid open with all the fimplicity and perfpicuity which the commentator could give it; and with as much particularity as the patience of the reader, a faculty rather apt to be discomposed by much exercife, could well be supposed to bear. CHAP. CHA P. I. Containing an explanation of the two first propofitions. THE two firft propofitions are thus explained by Dr. Hooke. "Euclid having premifed his principles, he begins his method 66 of demonftration, in which he takes no more for granted than "what he hath already laid down as easy and self evident. His first propofition then is upon a right line given to make an equilateral triangle. He hath defined in the fourth definition what he " means by a right line, namely that which lieth ftraight between the two extremes of it which are points; and what he means by an equilateral triangle, namely, fuch a one which hath all its "three fides equal to one another.' "This first propofition is a problem, which explains a way how"to do and perform the thing required, as well as fhews how to "manifeft the truth and certainty of the thing done: It contains "therefore and shews a double invention, without which or some' "fuch other thing, the propofition can neither be done nor demon"strated; which inventions are called mediums of means by which "we attain to the end propounded or defired. The end here fought is how from the ends of a line given to draw two other "lines, each equal to the given line, which fhall meet in one and "the fame point. It is certain that thefe lines muft begin from "the ends of the first line given; but which of these to draw first, " and which way, and with what inclination to the former line, "that is with what angle, that is not yet known, and fome inven"tion must be thought of, how to direct our ruler to draw it. "Well how shall this be done, fince there may be infinite of "lines drawn from each of thole points, which shall every one of "them be equal to this line given? How then fhall we among "those infinite or indefinite number chufe out the right? 'tis im"poffible without fome invention. Our author therefore helps you to one, and one which you have already granted to be feaf"ble in the third petition. Upon the center A, and distance AB, "draw a circle, fays he, BCD; what then? To what purpose; Why "Why this circle then will give you a line, in which are contained C "or means to perform the problems required to be done, which · "as AB, to find a point as C, to which lines being drawn from "the points A and B, they shall each of them be equal to one "another, and to the line AB which is given: or two points A and B being given to find a third point as C, which fhall have "the fame diftance from A and from B that they have from one * another. But our author not having given any definition of diftance or equality, otherwise than may be collected from equality "of the fides of fome figure; or of the rays or lines drawn from the center to the circumference of a circle, he chufes rather to "make use of an equilateral triangle to find out that propriety of a point fo pofited." "The fecond part then of the propofition is the demonstrative "part thereof, namely, to prove from the principles already laid "down and granted and affented unto for true and certain, by a “clear chain of reasoning and deduction, that these lines AC and BC are each of them equal to AB and fo equal to one another; "and confequently that the figure ABC bounded and limited by "them is an equilateral triangle, according to the description of "that figure in the twenty fourth definition. The next thing then "to be invented or found out is the medium or means of demonftrating it to be fuch; for this we have two. First, the definition "of the properties of the lines from the center of a circle to the "circumference in the fifteenth definition, that they are all equal "to one another. And fecondly we have the firft axiome, thofe "which are equal to one other are equal to one another. Firft AB " is equal to AC, because they are right lines drawn from the center "A to the circumference BCD; for by the fifteenth definition, as “I said, all such lines must be equal. Next BA and BC must upon the fame account be equal to one another, because they "alfo are lines drawn from the center B to the circumference "ACE: Therefore both AC and BC are equal to AB; but by the "first axiome those which are equal to one other are equal to one "another, therefore AC and BC are alfo equal to one another. "Therefore the three fides of the figure ABC; namely, AB, AC, "and BC are equal to one another, and confequently bound an "equilateral triangle according to the twenty fourth definition, "therefore |