him, by mistaking its ufe, fhould become the occafion of feveral prejudices. But to be more particular, Let ABC and DEF be two triangles; having the two B CE F This now is the hypothefis or fuppofition, which is to be confidered as abfolutely certain; and a principle from which we are to reason as confidently, as from the first common notion that magnitudes, equal to the fame, are equal to one another. Whatever might be collected from looking at the triangles, or by any other means, is to be entirely neglected; and not merely neglected, but even fhunned as leading to the most pernicious errors. The ftudent has only to examine the hypothefis, one part after another, making use of the figure only to affift him in comprehending its meaning, which he might set about in fome fuch manner as this, repeating to himself, the two fides I find are equal, but not any how; for they are faid to be equal each to each; the fides of all triangles are inclined to one another; but here a particular inclination is specified; they are faid to be equally inclined: this the attentive reader will comprehend perfectly, and be able to say I understand now what the author means; he affirms that all triangles which agree with one another in the circumftances above mentioned cannot poffibly fail in giving us the fame confequences. But let us see what confequences he fays will follow from his suppositions— They will also have the third fide or base BC equal to the base EF; and the whole triangle ABC equal to the whole triangle DEF : and the remaining angles top, why that epithet remaining? Why, because only one angle in each triangle being equal by the fuppofition, are there not then two in each triangle which remain to be confidered? But to proceed, the remaining angles fhall be equal to the remaining angles not both taken together, but each - to each. EACH TO EACH, this though somewhat particular nevertheless admits of a latitude fufficient to perplex an ordinary underftanding; and without fome additional circumftance, must remain unintelligible even to the acuteft. But the angles are particularly defcribed and also fixed; because they are to have the lines which are equal by the fuppofition extended under them; that is the angles under which AB and DE are extended are to be equal, as alfo thofe under which AC and DF are extended; viz. the angle ACB or BCA equal to the angle DFE or EFD; and the angle ABC or CBA equal to DEF or FED. Now these are said to be the confequences which will certainly follow from the forementioned fuppofitions. But it will be too much for the learner to attempt the whole demonstration at once; let us therefore see what confequence will follow from each part of the fuppofition taken fingly. The principle by which they are to be examined is the eighth common notion, magnitudes which apply themselves exactly to one another are equal. It is here faid that AB and DE are equal. What do you understand by this word equal? Certainly it must mean, that, if the straight lines AB and DE be properly placed, they will apply themselves exactly to one another. But what is the proper way of placing them? Put the point A upon the point D and the line AB upon DE; obferve all this may be done whether the lines be equal or unequal; but then the point B will apply itself to the point E only when the lines are equal; for when AB is longer than DE the point B will be found beyond E; and when AB is fhorter than DE the point B will be found between D and E. After a careful examination of these two particular lines; let us fuppofe each of them a thousand miles in length; here our fenfes forfake us, but furely our understanding unless it be weak indeed, will give us as pofitive a conclufion as in the other case; so that we may conclude univerfally that whatever be the length of AB provided only AB and DE be equal; and the point A upon the point D, and the line AB upon DE; the point B will always apply itself to the point E; which was the first consequence to be examined. Also if the point A be upon D and the line AC upon DF; the point C will apply itself to the point F, as is obvious for for the fame reasons; and that will always be the cafe whatever be the length of AC. We come next to confider in what circumstances it is poffible. to place AB upon DE; and at the fame time AC upon DF: Now this can always be done, if the angle BAC be equal to the angle EDF; because when the point A is upon the point D: and AB upon DE; it follows from the equality of the angles that AC will take the direction of DF. But if the angles be unequal, although the point A be upon the point D; and AB upon DE; yet AC cannot poffibly fall upon DF; because AC will then take a direction, on the one fide or the other of DF, according as the angle BAC is greater or less than EDF. 8 Laftly let us fuppofe the two extremities of two straight lines to be the fame; may we not conclude from this that the lines are equal, and in the fame direction, that is the one line will apply. itself to the other; because if it did not the two ftraight lines. would inclose a space. Therefore in this figure of ours, if it can be proved that the point B applies itself to E, at the fame time that the point C applies itself to F; we may certainly conclude that BC will apply itself to EF and be equal to it. Each of these fuppofitions ought to be examined frequently, and the confequences which follow from them are to be strongly impreffed upon the memory; fo that the very mention of the fuppo-fition may fuggeft the confequence.. And here it may be proper to inform the ftudent, that he is not to confider this as any acquifition of scientific knowledge; but only the confequences which common good fenfe cannot fail to draw, whenever these things become the subject of confideration; thefe remarks are introduced not as teaching any thing new, but only to fix the attention. However the combination of these, as in the fourth propofition, will carry us a step beyond the common fense of mankind: for he who first found out that the two fides and included angle fix every part of the triangle beyond a poffibility of change, had certainly more than a common notion of a triangle, and merited the high title of an inventor. The learner is now to endeavour to make himself master of the fourth propo-sition, before he proceeds to the next chapter, in which he will. find fome remarks tending to correct, confirm or enlarge the notions which he may derive from it according to the different degrees of attention bestowed upon it. THE reader is without doubt furprized to find this affair of fuppofitions represented as a difficulty almost unfurmountable; and not a little inquifitive, we may fuppofe, after the reafons of this ftrange phænomenon. Our own indolence is the real caufe, for fimple reafoning is hardly fufficient to fet the mind in motion without fome external application. When a problem is propofed, there is fomething to be done, and, which is very much to our purpose, may be done wrong: this fets us upon thinking; and by giving a preference to one method above another, we come to diftinguish between what is right and wrong, and become so much interested as to give the question a serious examination. But the cafe is very different in theorems, for the confequences are abfolutely fixt by the suppofitions, so that a mind without experience of such subjects has nothing to engage its attention. It would be abfurd to propofe a problem of this kind: Suppofe two triangles to have two fides equal to two fides, each to each ; and the angles, contained by the equal fides, equal; it is required to make the third fide, equal to the third fide; and the two remaining angles equal to the two remaining angles, each to each; namely those under which the equal fides are extended. Here there is no room for a construction; because according to the fuppofition, the fides and angles cannot be otherwise than equal. However it is neceffary to fhew that these magnitudes are really equal ; and the suppofition alone is what we must truft to, for bringing this about and we are at a loss because, for this end, we begin to derive the confequences from the fuppofition before we have examined the different circumftances of which it confifts, or without understanding what isimplied in the fuppofition, when it reaches beyond the obvious meaning of the words in which it is expreft; expreft; or when more ismeant than directly meets the ear. The mind must have room to exercise itself in, as well as the body; and it is rendered almoft inactive by being confined to a fingle fuppofition. But if the student has patience this may be remedied by varying the fuppofition; and then you may have all the latitude and range of thought, which is allowed in the conftruction of a problem. Let us fuppofe then that only AB is equal to DE, and the angle BAC equal to the angle EDF; but that we are quite in the dark as to AC and DF. When the point A is put upon D, and AB upon DE the point B will apply itself to E as before and AC will take the direction of DF; but it is impoffible to determine any thing concerning the pofition of the point C, only that it will be found fome where in the line DF or in DF produced. Again let us fuppofe only the two fides equal; then the point A being put upon D; and AB upon DE; the point B will as before apply itself to E, but we can make no use of the other part of the fuppofition; for we can determiné nothing concerning the direction which AC may take, because nothing is faid concerning the angles or inclinations of the lines. But farther, let us fuppofe, that not only AB is equal to DE and AC to DF and the angle BAC equal to the angle EDF; but moreover, that AB is equal to AC and confequently DE to DF: now by placing the triangles as in the propofition, we fhall find the angle ABC equal to DEF; and not only that, but, by putting A upon D, and AB upon DF the point B will now apply itself to F because AB and DF are equal by the fuppofition; and also the angle ABC will be equal to the angle DFE: but ABC has been already proved equal to DEF; therefore by the first common. notion the angle DEF is equal to DFE: that is the angles at the bafe of the triangle DEF are equal: But this conclufion does not depend entirely upon DE's being equal to DF; but it is required befides that BA fhould be equal to AC and to each of the lines DE, DF and likewise the angle BAC equal to EDF. So that this cafe can by no means be confidered as including the fifth propofition, in which there is no other fuppofition but the equality of |