them. And this may be acquired in fome fuch manner as the following.

Let the ftudent provide himself with a ruler and compaffes, and after fome practice in drawing ftraight lines, and defcribing circles; he is next to proceed to the examination of the common notions, as if they were properties of ftraight lines only, and true of nothing else. For without this precaution he will undoubtedly be liable to have the diftich quoted in the last chapter applied to him. And any tincture of the budibraftic genius difqualifies a man for this science; and excludes him from a great deal of rational amusement, to fay nothing of more folid advantages. I fhall therefore at the porch, not only lend the learner my advice but also my affiftance in ftriping himself of those prejudices which would difgrace his behaviour after he has been admitted into this magnificent temple where all the wonders of the world are displayed.

The reader may believe that I never would have introduced this advice with so much form and circumftance, without a firm perfwasion that it is of the last importance. He is therefore immediately to fet about the work, by defcribing a circle, not a geometrical but a mechanical circle; and fuch as any ordinary compaffes will exhibit; drawing at the fame time feveral ftraight lines from the center to the circumference. He is next to fatisfy himfelf of the equality of these ftraight lines, by measuring them with his compaffes his conclufion will be, that they are equal; and he will find his opinion of their equality grounded upon the first common notion; because they are all equal to the fame length, viz. the distance between the extreme points of his compaffes. But it is carefully to be obferved that this is not to be made the fubject of a tranfient reflexion, but of frequent and close meditation; varying the center and radius to the utmost limits of the compaffes; with now and then a thought upon the limited nature and impérfection of the inftruments.

The second and third of the common notions may be examined by defcribing two circles with the fame center, but at different distances, and drawing ftraight lines from the center to the remotest circumference; the parts of the ftraight lines intercepted between the two circumferences are equal; and will illustrate the

fecond

Ce

fecond common notion by taking the less radius from the greater. And thus we are to proceed untill we have fatisfied ourselves that thefe common notions are true at least of such straight lines as we can draw upon a piece of paper.

I beg the reader's pardon for my impertinence; but he is farther to be admonished, that it is not fufficient to run these things over in his own mind; but that he must be able to express them to the conviction of a by stander; and this will make it neceffary to dif tinguish his lines and circles by the letters of the aiphabet.

CHA P. III.

The fame fubject continued.

SUPPOSING this business of the straight lines accurately difcuffed; the learner is next to shut his compaffes; and then observe their progrefs in opening until they take the direction of a straight line: during this operation, he will find the inclination of the legs continually varying at first nothing, then gradually increasing until it disappears when the legs become one straight line. This inclination is a quantity, though not a tangible substance, but this the reader will do well to convince himself of; and for this purpose he may observe that any particular inclination may be equal to another, or the half or the third part of it. But the common notion of this kind of quantity is not so regular or determinate as that of a ftraight line; though it exhibits every poffible fhape which it can take in opening the compaffes as above directed: the reader therefore will be pleased to instruct himself properly in this and then proceed to examine whether the common notions are not also true when applied to this kind of quantity.

And for this purpose I would recommend a triangular piece of wood, of the shape of a right angled triangle with unequal fides, being afraid to meddle with circular arches, leaft we should conjure up a prejudice which we might want art afterwards to lay. By the affistance of this triangular piece of wood, make two equal inclinations (or angles) upon paper, taking care to make the lines unequal, to prevent prejudice. After these are made, their equality

may

B

A

C

E

G

may be inferred from the first common notion, as each of them will be equal to the inclination of the two fides of the peice of wood add to these two equal angles, other two equal angles; which may be done by the affiftance of a different corner of the fame piece of wood; and this will illuftrate the fecond and third; according as you confider one of them as taken away from the whole angle made up of the two; or as added together to make one. But it will be neceffary previous to this, to acquire a ready and accurate way of expreffing the different inclinations of lines, (called angles) by the letters of the alphabet. The figure annexed will be a very proper one for practice and the task which I would fet the reader is to tell the number of angles and the different methods of expreffing them; giving him to understand F that their number is above fourteen; and that, CAB, CAF, CAD; GAB, GAF, GAD; D ·EAB, EAF, EAD; BAC, BAG, BAE; FAC, FAG, FAE; DAC, DAG, DAE; are only fo many different ways of expreffing the fame angle, nor does this great variety, in the least puzzle or perplex the conceptions of an adept. This looks fo much like a riddle that I think it cannot fail to engage the attention of the curious. But not to truft entirely to the reader's own ingenuity for unraveling this knotty point; let him obferve the following hints; the letter at the meeting of the lines, whose inclination to one another we want to exprefs, is put in the middle, and it is fufficient that the other two letters, each express fome point in each line: thus the inclination of FB to BC is called the angle FBC or CBF and the inclination of DB to BC is the very fame with the other, as is obvious, and is called the angle DBC or CBD the inclination of BC to CE is called the angle BCE or ECB; and the inclination of GC to CB is the fame with the other and is called the angle GCB or BCG. But farther the angle ABG is made up of two angles viz. ABC, CBG: and the angle ACF is made up of two angles viz. ACB, BCF. And to affift the reader in applying the second common notion I have made the angle ABG

equal

сед

equal to the angle ACF: and I have likewise made the angle CBG equal to the angle BCF; and the conclufion will be that the angle ABC is equal to ACB.

CHAP. IV.

In what manner our common notions begin to take a fcientific form.

AFTER the reader has prepared himself according to the directions given in the last two chapters; it will now be proper to take a review of the inftruments, which he made ufe of, for regulating his conceptions: and thefe, he will find, were very limited, being confined to a few inches. Let him next afk himself, whether he has any reason to fufpect, that the conclufions, obtained by the

Huse one
the
most impor:
tant princ. help of these inftruments, were equally limited. Nor is this point
July to be secuto be determined rafhly, but so as to be still certain that he treads
from Ma: on firm ground; and we may venture to draw this conclufion for
thematicks him; that, without any great force upon his imagination, he can
conceive his inftruments to have a double or triple extent without
finding the least reason to change his opinions. And by proceeding
thus, he will certainly come to this conclufion at last; that al-
though these instruments might be the occasion of his turning his
thoughts to this fubject, yet his opinions were nevertheless derived
from the nature of extenfion in general; and that they knew no
other limits, but fuch as bounded extenfion itself: but more par-
ticularly, that a circle whofe radius is a thousand miles, or the
thousand part of an inch, would furnish the fame conclufions as one
of two or three inches. Here now our opinions, which before
were measured by our inftruments, begin to put on a different form
and display to our imagination the firft dawn of science.

If any one should pretend that he had the notions orginally in this very general form to which I have been endeavouring to lead him; I have only to say, unless they were acquired by an examination of particulars, he will find his notions fit every thing fo well, that when he comes to apply them to particular instances, he will not be able to tell which is which.

The reader is to endeavour next to get fomething like a scientific notion of an angle, by correcting the vulgar notion of an angle,

by

by which is understood the corner of any thing. Now this does not so much depend upon any stretch of the imagination, by which large objects, and such as exceed the experience of our senses, are to be made the subject of Contemplation; because the point where the lines meet, together with any point in each of the lines fixes the angle invariably or in other words, the three points denoted by the three letters of the alphabet, expreffing the angle, fixes any rectilineal angle: for the angle is not changed by making the lines longer or fhorter; but only by opening or shutting them; conceiving them to turn upon a pin like the two legs of a pair of compaffes.

But our inftruments are not only too limited for our conceptions, but are inaccurate in other refpects. We have a very clear notion of three dimenfions viz. length, breadth and thickness and surely without nicely separating and distinguishing these, it is impoffible to have true and proper conceptions of magnitude. But these different dimenfions cannot be represented by our inftruments. For when we attempt to draw a line or even to mark a point; our line and point poffefs all the three dimensions in as great perfection as a cannon ball or the maft of a fhip. The human mind, when once made fenfible of its powers, will never fuffer its conceptions to be fo cloged with matter: which has put those who carry their views beyond the vulgar, upon inventing fome method by which our conceptions may be rendered more rational and confiftent; and this is the original of definitions.

CHA P. V.
Of definitions.

OUR author has proceeded with fingular judgement in laying down his principles where the common notions are fufficiently distinct and accurate, he has inviolably adhered to them. But when these are too incorrect or too indeterminate, he explains the fenfe in which he would have any particular term be understood; and what conception he requires his reader to have of the figures which he defines. Definitions may be confidered as of two kinds; firft, fuch as ferve only to explain the meaning of a word; but these VOL. I.

b

are