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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 some practice in drawing ftraight lines, and describing 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 amufement, to say nothing of more folid advantages. I fhall therefore at the porch, not only lend the learner my advice but also my affistance in striping 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 fo much form and circumftance, without a firm perfwafion that it is of the laft importance. He is therefore immediately to set 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 straight lines from the center to the circumference. He is next to fatisfy himself 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 clofe 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 remoteft circumference; the parts of the ftraight lines intercepted between the two circumferences are equal; and will illuftrate the fecond
fecond common notion by taking the lefs 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 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 diftinguish his lines and circles by the letters of the alphabet.
CHA P. III.
The fame Subject continued.
SUPPOSING this business of the straight lines accurately discusfed; the learner is next to shut his compaffes; 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 fo regular or determinate as that of a ftraight line; though it exhibits every poffible shape 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, least 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 thefe are made, their equality
may be inferred from the first common notion, as each of them
equal to the angle ACF: and I have likewife made the angle CBG equal to the angle BCF; and the conclufion will be that the angle ABC is equal to ACB.
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 These one the confined to a few inches. Let him next afk himself, whether he has any reason to fufpect, that the conclufions, obtained by the most impor: tant une help of these inftruments, were equally limited. Nor is this point ply to be sea to be determined rafhly, but so as to be still certain that he treads ced from Ma. on firm ground; and we may venture to draw this conclufion for force upon his imagination, he can thematicks him; that, without any great 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 although these instruments might be the occafion of his turning his thoughts to this subject, 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 particularly, 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 difplay 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 fay, 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 inftances, 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 ftretch 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 shorter; 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 thefe different dimenfions cannot be reprefented 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 dimenfions in as great perfection as a cannon ball or the maft of a ship. 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.
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 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 confidered as of two kinds; first, fuch as ferve only to explain the meaning of a word; but these VOL. I.