"therefore upon the line AB given, an equilateral triangle is made, "which was the thing to be done and proved."

"THE second propofition is alfo a problem.

"From a point given to draw a right line, which shall be equal "to another right line given. For the performing of this propo"fition, it being a problem, there is need of two mediums to be "invented or found out; the first is for the doing of the thing re

quired; and the second for the demonftration of it; and both "these are to be fetcht out of the principles already agreed to, or "from the truth evidenced in the preceding propofition: For we "are not to suppose any thing further known in this fcience, and "therefore much less are we to make use of it. Searching there"fore our store, we have no other medium to make a line equal to “a line, than first by the help of a circle, defined definition 15. "which by the third poftulatum is granted to be defcribable upon any center and at any distance: Or fecondly, by an equilateral "triangle described by definition 24, which we learnt how to make "by the preceding problem: For as to the equal fides of an isof"cles triangle, definition 25, or the equal fides of a square, a pa"rallelogram or oblong fquare, Rhombus or Rhomboeides defcri"bed definition the 30, 31, 32, and 33. Though their properties are there defined, yet we are not taught how to make them as "yet, and confequently can make no use of them, as media to perform the thing required to be done by this problem. Nor " are we to fuppofe, that the length of the line given may be taken by the help of a measure or a pair of compaffes, and transferred "to the point given; because those are first not mentioned in the principles laid down there, which you are to make use of, and "of no other, till they be accepted for principles undeniable: For "this is not yet granted, that you can with compasses, take a true length of a line, much lefs that you can transfer it and set it in "another place. But you have granted that 'tis poffible, upon a point given at any diftance, to defcribe a circle, or fuppofe it fo "done, which is fufficient for the demonstration, that being the principal thing aimed at by our author, namely, to lay open to

66

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« the understanding the refons and grounds of the properties of /d "quantities fo and fo qualified, that you may plainly fee how and

VOL. I.

d

for

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6.6

66

"for what caufe things are thus or thus and cannot be otherwise. "For as to the most practicable and expedite ways of doing those "things mechanically, and for other ufes, that belongs to another part of mathematics; namely to the practical part thereof, which "is called practical geometry, or mechanical geometry, which ought not to be learned till this be first known; but this which "our author treats of is fpeculative geometry, and principally aims "at demonstrations or explaining the proprieties of quantities to "the understanding. You faw clearly by the former propofition why ABC was an equilateral triangle; and there could be but "two fuch made upon the line AB in the fame plain; there being "but two points where in those circles cut each other; thofe cir"cles determining all the lines equal to AB that can be drawn "from the points A and B. His way then of performing this "problem is this: Let the right line given be BC and the point 66 given be A; from "which point a right "line is to be drawn 66 equal to the line BC. "First draw a line from "B to A, which is "granted poffible by "the first poftulatum ; "then by the former propofition upon this "line,BA make an equi"lateral triangle BAD: "then upon the center "B and distance BC "defcribe a circle, as "CGH by the third postulatum; then by

K

H

D

A

L

E

B

"the second poftulatum produce DB to F; then upon the center "D and distance DG draw the circle GKL, then as before, pro"duce the line DA to E; there shall AL be the line required to "be drawn from the point A equal to the given line BC."

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"This is the construction of the problem, or the preparing of "the propofition fit for demonstration, by which you may clearly "understand the reafons of it, deduced from the few principles "already laid down: For first, that BC is equal to BG is clear "from the fifteenth definition, which determines the propriety of equality of the rays of a circle. Next that DBG is equal to "DAL, is as clear from the fame definition, they being both rays "or lines drawn from the center D to the circumference GKL by "the construction premised. Thirdly that DB is equal to DA, is "clear from the conftruction; for DBA is an equilateral triangle, "two of whose fides DB and DA are. Now by the third axiom "or common notion, if from equal quantities you take equal quan"tities, the remainders fhall be equal; if from DL you take DA; " and from DG, DB; the remainder AL fhall be equal to the re"mainder BG; but BC is alfo by the construction equal to BG; "therefore fince by the firft axiom these two quantities which are "equal to one other quantity are equal to one another; therefore "BC and AL, being equal to BG, are equal also to one another; "therefore from the point A the line AL is drawn equal to the "line BC; which was the thing to be done and proved."

"Now though this way of demonstration and reasoning may "seem tedious and too long to detain the mind and attention in "the finding out the proprieties of quantities, yet 'twas the way "made ufe of by the ancients. And 'tis altogether neceffary, efpe

cially in the beginning of this ftudy, to accuftom the mind to "attention and circumfpection, that it may receive nothing for "truth but what it fees clearly by the reasons and causes of it, "that thereby the mind may acquire an habit of intention, and " of examining the whole chain of confequents from the first principles to the truth evidenced. For the want of which, fome "fmall error perhaps may flip into the mind under the appearance "of truth, and thereby make all the fubfequent reafonings and "deductions unfound; and 'tis very much harder to clear and free "the mind from it when once received, than to prevent the recep"tion thereof. There cannot therefore (in this study especially, "not now to mention any other, where it is poffible it may be altogether as convenient, nay neceffary) there cannot I fay

"therefore be as I conceive toomuch circumfpection and caution "used in admitting principles, and furnishing the mind with the "true grounds of knowledge; because for the most part we are too 66 prone to take up every thing we hear upon truft, without exami"nation: we are too apt to run away with a thing, and think we "know it and fee it clearly before we are sure we do; and are "impatient of delay in examining and confidering; whereas if the "mind be a little at firft accustomed to this leifurely and strict way of reasoning, after it has got a habit it will make as much dispatch in receiving things with sufficient examination, as ano"ther shall without it. And the patience only is needful for the "most part at first to beget attention; nor is it peculiar to this "acquifition alone; but we fee it neceffary, and practised in many "other things where a good habit is to be acquired; as in reading,

writing, mufic, drawing, and most other manual operations. "The roots and beginnings of knowledge, and practice too, are "bitter and tedious, but the fruits are sweet and pleasant; and "whofoever attains the end, will never repent the time spent in "the beginning."

СНАР. II.

The fame fubject continued.

IN the preceding chapter I have given Dr. Hooke's explanation of the two first propofitions, in order to fhew the reader that I am not fingular in my opinion, that Euclid is not to be understood, unless the learner bestows that attention upon the first principles, which may enable him to carry along with him their full meaning and import. It is true that some confused facts concerning the properties of figures may be picked up by a very careless perusal of this author: but whoever is fatisfied with this, had better look for his knowledge fome where else; because he will meet with a great many impertinent interuptions to his scheme from the several steps of the demonstration; and rather content himself with a kind of law evidence, by refolving to confent to every proposition which is delivered as truth by two or more credible authors.

Although

Although I approve in general of what Dr. Hooke has advanced, yet I am convinced that a more minute confideration of the different parts of each propofition will be neceffary for the ftudent who would wish to leave his prejudices behind him as he advances. I shall therefore point out these circumstances, which he might be apt either to overlook or mistake. Suppofing the first propofition of Euclid carefully examined, he will find it taken for granted that the two circles cut one another, and that this supposes the circles to be described upon the fame even surface, or according to the geometrical language in the fame plane, otherwife there could be no fuch point as C. Again the fuppofing C to be a point implies that the two lines, the two circumferences of the circles have no breadth, otherwise the point would have parts. Likewife by fuppofing the triangle ABC to be an affignable or determinate magnitude implies that the points A, B, C have no magnitude for if they had parts various triangles differing both in fides and angles. might be described, and it would be impoffible ever to arrive at a determinate conclufion.

But farther, the reader is to confider that he has been reasoning all this while, upon a particular straight line of very inconfiderable length; He is therefore next to inquire, whether his constructions and conclufions are alfo particular. That they are not will appear from this, that no confequence is fuppofed to follow from the ftraight lines having any particular length, nor is any construction undertaken upon that fuppofition; only that the line be finite that is, that we know its two ends.

But it happens rather unluckily, that though the scientific conftruction be general we are nevertheless forced to take up with a particular one; which is very apt to create prejudices unless our attention be every now and then called to the general construction: we must remember that, though we work with a ruler and compaffes, the science knows no fuch inftruments. Let us fuppofe the line AB ten miles in length; and that the same construction is to be performed; we shall now get beyond the objects of our senses; but if we have bestowed the necessary attention to the problem, the understanding will have as clear and diftinct a perception as it had before, when the line was perhaps not above three inches long.

Our