are not properly geometrical definitions, for from this no confequence can follow unless there be a myftical virtue in the name. Secondly, thofe from which confequences or properties do follow, which may be called geometrical to diftinguifh them from the others. The definition of a circle, given by Euclid, is of this kind for all the properties of a circle follow as confequences from it. And here the reader is to be admonished, that, upon this and fuch like occafions, his common notion of a circle is to be laid afide, and nothing admitted as a property of this figure, unless it can be fhewn to follow from this definition.

Though Euclid in the arrangement of his principles has placed the common notions after the definitions, yet they are prior to them in the order of conception; and indeed if this is not attended to, fome of his definitions will be unintelligible; for inftance his definition of a straight line. He fays a ftraight line is that which lies evenly to the points in itself. Now if I am to conceive nothing previous to this, refpecting a ftraight line; what can I understand by this definition; or what can I infer from it? The reader will be just as much at a loss to conceive the meaning of the word evenly, as of the straight line itself. But if we confider this definition as an improvement upon the common notion of a ftraight line, (fee com. not. 12.) every thing is very intelligible for after a proper examination of this principle, that two ftraight cannot inclofe a fpace; every body will infer, though not scientifically neverthelefs very confidentally, that every straight line muft lie evenly to all the points in itself; otherwife he certainly might have hopes at least of making two of them inclose a space. I would be rightly understood upon this point; nobody can imagine that it is my opinion, that Euclid intended that the one of these should be inferred from the other fcientifically; but only that the definition expreffes the conception, derived from two lines, reduced into a more fimple form; though indeed he himself reafons from the common notion as will appear in the fourth propofition.

Dr. Hooke one of the greatest improvers of practical mathematics, was notwithstanding fo fenfible of the neceffity of laying an accurate foundation in theory that he thus fpeaks concerning a point. A point is that which has no part. This which fome

"would

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"would deem the most inconfiderable thing in the world, feems yet the most difficult to be understood; no fenfe, or imagination. "or fant'cy can reach it, nor words defcribe it, but by a negative, "to tell you what it is not: For it is not to be taken in the sense, “that the whole Earth is called a point in respect of the Universe, "nór in the sense that the end of a tapering thing is called a point, as of a pin or needle; though they seem to be the smallest things we know; because these latter may be said to have as "many parts as the fore-mentioned; for fince all quantity is divi"fible in infinitum, the least quantity, may be divided as often as "the greatest, and therefore whatsoever is divifible must have parts, and therefore none of these can be properly called a point, "in the fense here named, unless this point be understood to be "the apex of a mathematical cone or pyramid, where the super"ficies of it is determined, for that will be a mathematical point: "But it cannot be fuppofed of a phyfical point, or material cone "or pyramid, for that will have extenfion and bluntnefs. And we "find that microscopes will make those points divisibe even to sense,

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nay even almost to discover a new world in a point, nay there is “one now that affirms he has seen more than ten thousand living "creatures in the bigness of a very small fand, which itself indeed " is but a visible point to the naked eye; and each of those ten "thousand may have worlds within them. We know not the “limits of quantity, matter, and body as to its divisibility or ex"tenfion; no imagination can comprehend the maximum or the "minimum naturæ, our faculties are finite and limited, and we "must content ourselves within the orb and sphere of their acti"vity. And acquiefe in a negative definition, and understand if "we can somewhat that is smaller than the smallest, though that "be alfo improper; for in that which is not quantity, there is "neither smaller nor bigger, we must endeavour to understand "some what infinitely little, less than which there cannot be, "fomewhat that has no bigness or extenfion or quantity, but only

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pofition and respect to quantities circumjacent: From which, to "this or that body, there is a determinate length and distance; "and upon this account, wherever we endeavour to understand this notion, our imagination will reprefent to us the smallest

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"vifible body, as an exceeding fine fand, or a mite, or the point "of a needle, or the smallest vifible body we have ever seen on paper, or the like; which we must be content with, fince the "fant'cy forms nothing but what is first in the sense, though it be "none of these. And in truth it can have no true definition that "will reach its effence. Analogous to this point, fign or nothing "in quantity is the nought, cifer or zero in number: The never "in time: The reft or quiet in motion. For as no aggregate of "points will ever produce a line or quantity; fo the multiplication "of noughts or cifers will never produce a number; and as the “addition of nevers cannot make time, fo the aggregate of rests "cannot produce motion. So that all these may not improperly "be called the terminus or bound, from which they all begin; fo quantity may be said to begin from a point or nothing: number may be faid to begin from nought, cifer or zero: time may be "faid to begin from never and motion to begin from reft. And "as the minimum naturæ may be faid to be the first quantity; if at "least there be a minimum in nature, fo an unite may express it in "numbers; inftant in time, and moment in velocity. It may poffibly be thought I have faid too much of nothing, but yet "feems to be of the greateft confideration in nature; for it seems "to be the beginning of every creature; even the greatest creatures having been traced to begin from an atom or point, no eye or "sense can reach it; nor any understanding limit it; that the be ginning of a very large animal hath been seen alive, ten thousand "times smaller than a mite, may be proved, and yet how much fmaller it may have been is not determined."

it

It has been said of the mufick of the spheres that it is so loud we cannot hear it and of a mathematical point, according to this explanation, it may be said, that it is fo fimple it is impoffible to comprehend it. This no doubt, muft prepoffes the reader with ftrange apprehenfions of difficulty in the profecution of a science which fets out fo unaccountably; requiring us to admit as a principle what we cannot comprehend.

I would not have thrown this metaphyfical bugbear in the way of the reader; only I was afraid left he might stumble upon it of himself, with fome hazard to his understanding, if he should pursue

the

the metaphyfical analysis of it, until he had no doubts left but that it was really incomprehenfible; especially if he happened at the fame time to discover, that the fame ingenious fubtilties would also apply to lines and furfaces, by which means they might likewise be entitled to a place among the incomprehenfibles.

I fhall therefore beg leave to call his attention to a method of explanation very different from this, but which arifes very naturally from what I mentioned before viz. that the definitions should be confidered as fubfequent to the common notions; and introduced merely as auxiliaries to them. And in the present instance, we have a very clear notion of three dimensions in magnitude; though we always find them fo connected that it is impoffible to separate them from one another. Let us therefore try to contemplate them together, but one after another. Every part of space that we confider has a shape; for instance the shape of a room: this space the mathematicians call a folid; that which limits it or gives it this shape is called a furface; which must be confidered as having only length and breadth; because if it had thickness alfo ; then it would not be the boundary of the folid, but a part of the folid itself. Therefore the proper definition of a furface will be, that which hath length and breadth. But farther this surface has a shape or is limited; that which limits it cannot have length and breadth; for then it would not be the limit of the furface but a part of it. Therefore the proper definition of a line is length without breadth. Lastly this line has its limits; which limit cannot have length, for then it would not be the limit of the line but a part of it. Therefore the proper definition of a point would be, that which belongs to magnitude, and has no parts, that is none of the common dimensions, length breadth or thickness. Thus it is eafy to conceive how thefe definitions of a point line and furface are deri ved from the common notions of magnitude..

AS the definitions which we have been explaining are never quoted or appealed to as tefts by Euclid, the learner might be

ready

ready to conclude that they were placed here for no other purpofe but to imprefs one with an idea of the difficulty of this fcience.

Though it be true that they are never ufed directly, yet by being placed where they are, they stop the mouths of impertinent critics, who would be ready to start objections which are immediately removed by these definitions.

In this chapter I intend to explain what is meant by a PLANE, a plane rectilineal angle, a right angle and an acute and obtufe angle; firft confidering what opinion every one might form of these, by examining thofe inftruments or circumftances which would fix his attention to this fubject.

To understand rightly Euclid's definition of a plane furface, it will be neceffary to have recourse to the former fuppofition, that the definitions are improvements upon our common notions: for inftance, let us fuppofe one at work, in the manner of a carpenter, to fatisfy himself that a furface was even, fuppofe the furface of a table: he would apply a ruler in all directions from fide to fide; and finding that it touched the table in every point and in every direction, he would conclude that the furface was even. And now we fhall find that the definition is nothing elfe, but giving the notion acquired by this operation, a regular form and sufficient

extent,

For Euclid's plane furface which is commonly called a PLANE is this furface of the table, endued with a scientifical evenness instead of a mechanical one; and instead of being confined to any fhape; of an unlimited extent. It is in or upon such a surface as this, that Eulid fuppofes all his lines and figures to be defcribed and drawn, in his firft fix books: and this is the more carefully to be obferved, because, though it is not mentioned in the demonstrations, yet there are many properties of lines and figures, which are not true, but upon the fuppofition that the lines are in the fame plane, as I fhall have occafion to obferve.

And the plane rectilineal angle is the inclination of two straight lines to one another in or upon fuch an even furface as this, which meet together but are not in the fame direction. They might be inclined to one another so that when produced they would interfect

or