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them. And this may be acquired in some fuch manner as the following
Let the student provide himself with a ruler and compasses, and after some practice in drawing straight lines, and describing circles; he is next to proceed to the examination of the common notions, as if they were properties of straight lines only, and true of nothing else. For without this precaution he will undoubtedly be liable to have the distich quoted in the last chapter applied to him. And any tincture of the budibrastic genius disqualifies a man for this science ; and excludes him from a great deal of rational amusement, to say nothing of more solid advantages. I shall therefore at the porch, not only lend the learner my advice but also my aslistance in friping 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 circumstance, without a firm perfwasion that it is of the last importance. He is therefore immediately to set about the work, by describing a circle, not a geometrical but a mechanical circle ; and fuch as any ordinary compasses will exhibit; drawing at the same time feveral straight lines from the center to the circumference. He is next to fatisfy himfelf of the equality of these straight lines, by measuring them with his compasses : his conclusion 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 same length, viz. the distance between the extreme points of his compasses. But it is carefully to be observed that this is not to be made the fubject of a transient reflexion, but of frequent and close meditation varying the center and radius to the utmost limits of the compasses ; with now and then a thought upon the limited nature and imperfection of the instruments.
The second and third of the common notions may be examined by describing two circles with the saine center, but at different distances, and drawing straight lines from the center to the remotest circumference; the parts of the straight lines intercepted between the two circumferences are equal ; and will illustrate the
fecond common notion by taking the less radius from the greater. And thus we are to proceed untill we have satisfied ourselves that these 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 sufficient 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 necessary to distinguish his lines and circles by the letters of the aiphabet.
SUPPOSING this business of the straight lines accurately discussed; the learner is next to shut his compasses; and then observe their progress 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 straight line ; though it exhibits every possible shape which it can take in opening the compasses 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 sides, being afraid to meddle with circular arches, least we should conjure up a prejudice which we might want art afterwards to lay. By the assistance 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 be inferred from the first common notion, as each of them will be equal to the inclination of the two sides of the peice of wood : add to these two equal angles, other two equal angles ; which
may be done by the assistance of a different corner of the fame piece of wood ; and this will illustrate the second and third; according as you consider 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 necessary
A previous to this, to acquire a ready and accurate way of expressing 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
с which I would set the reader is to tell the number of angles and the different methods of expressing them; giving him to understand 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 so many different ways of expressing the fame angle ; nor does this great variety, in the least puzzle or perplex the conceptions of an adept. This looks so much like a riddle that I think it cannot fail to engage the attention of the curious. But not to trust entirely to the reader's own ingenuity for unraveling this knotty point'; let him observe the following hints; the letter at the meeting of the lines, whose inclination to one another we want to express, is put in the middle, and it is sufficient that the other two letters, each express some 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 same 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 assist the reader in applying the second common notion I have made the angle ABG
equal equal to the angle ACF: and I have likewise made the angle CBG equal to the angle BCF; and the conclusion will be that the angle ABC is equal to ACB,
In what manner our common notions begin to take a scientific 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 instruments, which he made use of, for regulating his conceptions : and these, he will find, were very limited, being confined to a few inches. Let him next ask himself, whether he
has any reason to suspect, that the conclusions, obtained by the font princi, help of these instruments, were equally limited. Nor is this point
from man firm ground; and we may venture to draw this.conclufion for Shimaticho. him; that without any great force upon his imagination, he can
conceive his instruments 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 conclusion at last; that although these instruments might be the occasion of his turning his thoughts to this subject, yet his opinions were nevertheless derived from the nature of extension in general ; and that they knew no other limits, but such as bounded extension itself: but more particularly, that a circle whose radius is a thousand miles, or the thoufand part of an inch, would furnith the same conclusions as one of two or three inches. Here now our opinions, which before were measured by our instruments, begin to put on a different form and display to our imagination the first 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 so 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 something like a scientific notion of an angle, by correcting the vulgar notion of an angle,
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, expressing the angle, fixes any rectilineal angle : for the angle is not changed by making the lines longer or shorter ; but only by opening or shutting them; conceiving them to turn upon a pin like the two legs of a pair of compasses.
But our instruments are not only too limited for our conceptions, but are inaccurate in other respects. We have a very clear notion of three dimensions viz. length, breadth and thickness : and surely without nicely separating and distinguishing these, it is impoflible to have true and proper conceptions of magnitude. But these different. dimensions cannot be represented by our instruments. For when we attempt to draw a line or even to mark a point; our line and point possess all the three dimensions in as great perfection as a cannon ball or the mast of a ship. The human mind, when once made sensible of its powers, will never suffer its conceptions to be so cloged with matter : which has put those who carry their views beyond the vulgar, upon inventing some method by which our conceptions may be rendered more rational and consistent; and this is the original of definitions.
CHA P. v.
OUR author has proceeded with fingular judgement in laying down his principles :, where the common notions are sufficiently distinct and accurate, he has inviolably adhered to them. But when these are too incorrect or too indeterminate, he explains the sense 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 considered as of two kinds ; first, such as serve only to explain the meaning of a word; but these VOL. 1.