Our author's plan obliges him ftrictly to prohibit the use of any particular inftruments; ftraight lines and circles are fuppofed to be drawn and described by mechanical operations; and we are left to guess in what manner. It will however be prudent to acquire fome one ready mechanic way at least of constructing every problem. For I always fufpect any pretenfions to general truths which have not been collected from an examination of particulars. The ruler and compaffes are fuch exact minature representations of the two postulates, that they are to be preferred to every other instrument for affifting the imagination to follow Euclid's conftructions: only the compaffes are to be confined to their proper ufe, always to defcribe circles, but never to measure diftances. Indeed it is not always neceffary, and often would introduce confufion if the circles were compleatly defcribed: it is therefore fufficient to defcribe as much of them as is neceffary; as for inftance, in the future application of this propofition, it is obvious that it will be fufficient to find the point C; because the point where the circumferences of the circles cut one another gives us every thing neceffary for the conftruction. We come now to the fecond problem which is of a more complicated nature than the first; and yet there are very few, upon reading it, who do not fall in with the notion that Euclid has taken a great deal of unneceffary pains to do what a single opening of the compaffes would have performed with equal or perhaps more exactness. But confidering the fcience only with a view to practice, the affairs of mankind are carried on upon a much larger fcale than to be managed with a ruler and compaffes; and we have only to suppose the line BC a mile in length, and this will answer the objection sufficiently. The understanding is capable of reaching the general conclufion delivered in this propofition; why then fhould it be fettered with inftruments? Euclid's contrivance is admirable and fuited to the dignity of the human mind. According to his plan we are guided by inftruments, but neither confined nor loaded with them. After the ftudent has fixed in his memory the conftruction and demonstration of this fecond propofition, it is more than probable that his conclufions will be limited to the particular figure which he he made use of and to that very position which the lines happened accidentally to take. This is a very general prejudice, and will be removed by attending to the following directions. Let the learner confider whether DA and DB produced will cut the circle CGH: and he will find that DB must cut it as it paffes through its center, but that DA's cutting it depends upon the length of AB. Change the position of the point A and repeat the conftruction and thus you will learn what lines have a fixt pofition or only an accidental one: describe the equilateral triangle ABD upon the other fide of the line AB and compleat the figure; alfo join the points A and C instead of the points A and B and repeat the same constructions; and by proceeding thus you will be able to conquer the prejudice of fenfe and to acquire fomething like a scientific view of the problem. For you will find no confequence deduced from BC's having any particular length or the point A any particular pofition. But the third propofition will afford an opportunity of taking a somewhat different view of these two propofitions. It is a problem. Two unequal straight lines being given to cut of a part from the greater equal to the lefs. The fimpleft cafe of this propofition would be, when both the lines are drawn from the fame point: for making that point the center, and the shorter line the radius, the circle so described would solve the problem: but if the lines had any other pofition, a very different apparatus would be neceffary; we must learn how to put a line at the extremity of the longer equal to the fhorter line; and this cannot be done before we have learned how to defcribe an equilateral triangle. I would therefore propose it as an exercise for the student, and his success would be a proof that he was master of the subject, to set out upon a supposition that the book began with this third propofition, which will now include the first and fecond; and by this process. he will have their connexion with each other ftrongly impreffed upon his mind. His figure will confift of five circles, and their use is to determine four points: fuch a point as C in the first propofition, G and L in the fecond; and E in the third. Whoever attempts to communicate knowledge to mankind, must write upon the supposition of a certain degree of improvement in those to whom he addreffes himself, otherwife his book can have no determinate end to answer. Euclid supposes his reader above the prejudice of fenfe, and to have the ready ufe of his understanding, with a due command of attention; and upon this supposition he has faid every thing neceffary to convey the fullest information to his reader. But I write to a different fort of people, to fuch as are immersed in the prejudices of fenfe, and at the fame time very thoughtless; to those whose understandings are not difpofed to attend to the call of reason without frequent admonitions; and this is my apology for begging the student's attention to one thing more, before I finish this chapter; he will always find something supposed or given in every propofition, and perhaps nothing will contribute fo much to a right understanding of the propofition, as a diligent enquiry into the use made of the data or suppofition, as for inftance whether the reasoning in the third propofition, does not depend upon the fuppofition that AB is longer than C; and where the reasoning would fail if that were not the cafe. Our author was very confident that his reader would know, that, without this part of the fuppofition, there never could have been fuch a point as E: but I am affraid that mine trusted to his fenses for the real existence of this very point; dont I fee, fays he, that the circle cuts it. I shall conclude this chapter by reminding the reader again, that the use of the two circles in the firft propofition is to find the point C: and in the second, that the equilateral triangle is described to fix the point D, which is to be the center of a future circle; whose radius DG is determined by the description of the circle CGH; and laftly that this circle itself is described to find the point L. And thus I leave the ftudent to pursue his own meditations upon these three propofitions, only advertifing him, that, if he thought them easy upon the first reading, and still perfifts in the opinion that he then understood them, I am certain he knows nothing of the matter. CHAP. /1 CHA P. III. Concerning hypothefes. THE fourth propofition is of a different kind from the three phir Aasile firft; and is called a theorem. In the problems fomething is re- If geometrical knowledge could be communicated in the form If two triangles have the two fides equal to the two fides, each to each; and have the angle equal to the angle, the angle, contaiVOL. I. e ned ned by the equal straight lines: They will also have the base equal to the bafe; and the triangle will be equal to the triangle; and the remaining angles will be equal to the remaining angles, under which the equal fides are extended, each to each. This general enunciation of the propofition will convey but a very indistinct perception of its meaning to a learner, especially upon a first reading; this however may be improved and rendered diftinct by the affiftance of a figure, reprefenting two triangles in the fuppofed circumstances: but here again a new difficulty arises, as this hardly ever fails to bring a prejudice along with it: for the triangles representing not only the fuppofitions, but at the fame time the inferences which are faid to follow from them; the ftudent is at a loss to distinguish what is given from that which is to be inferred from it; because in all probability he will look upon the figure itself as the only fource from which his knowledge is to be derived; and then he is as well convinced that the conclufions are true as the fuppofitions; and cannot conceive what it is he has to demonftrate. And unless he be qualified to lay a proper stress upon the fuppofition, the demonstration must appear to him an idle abuse of words calculated only to perplex his understanding, efpecially if he has already fixt his opinion by his compaffes and other inftruments. It is not eafy to devise a ready remedy for this error; fo that a man runs a rifque either of having no opinion concerning what is propofed in the propofition or an abfurd one; for of all the difficulties attending the acquifition of the fcience, this is the hardest to be got over. To reafon accurately from a fuppofition is no eafy matter; attention and habit will do a great deal, but above all a proper fenfe of the difficulty of it. In the problems which we have already confidered, the student has only to attend to the works of his own hands, in the first propofition the ftraight line AB is given him; but the two circles, and the two ftraight lines AC and BC are his own manufacture; and likewise in the fecond fo is the whole figure except the point A, and the ftraight line BC this makes a distinction which nobody is fo thoughtless as to overlook. But in a theorem like this fourth propofition; where one has nothing to do but to think, the cafe is very different, especially if the figure, which is intended to direct . |