:" Book V. ceffity of indirect demonstrations is avoided. In the whole of geometry, I know not that any happier invention is to be found; and it is worth remarking, that Euclid appears in an other of his works to have availed himself of the idea of indefinitude with the same success, viz. in his books of Porisms, which, have been restored by Dr Simfon, and in which the whole analysis turned on that idea, as I have thewn at length, in the third vohime of the Transactions of the Royal Society of Edinburgh. The investigations of those propositions were founded entirely on the principle of certain magnitudes admitting of innumerable values; and the methods of reasoning concerning them seem to have been extremely fimilar to those employed in the fifth of the Elements. It is curious to remark this analogy between the different works of the fame author; and to confider, that the skill, in the conduct of this very refined and ingenious method, which Euclid had acquired in treating the properties of proportionals, may have enabled him to fucceed so well in treating the still more difficult subject of Porisms. With such an opinion of Euclid's manner of treating proportion, as I have now expressed, it was impofible that I should attempt to change any thing in the principle of his demonstrations. I have only fought to improve the language of them, by introducing a concise mode of expreffion, of the same nature with that which we use in arithmetic, and in algebra. Ordinary language conveys the ideas of the different operations supposed to be performed in thefe demonstrations so slowly, and breaks them down into so many parts, that they make not a sufficient impreffion on the understanding. This, indeed, will generally happen when the things treated of are not represented to the senses by Diagrams, as they cannot be when we reason concerning magnitude in general, as in this part of the Elements. Here we ought certainly to adopt the language of arithmetic or algebra, which, by its shortness, and the rapidity with which it places objects before us, makes up in the best manner poffible for being merely a conventional language, and using symbols that have no resemblance to the things expreffed by them. Such a language, therefore, I have endeavoured to introduce here; and, I am convinced, that if it shall be found an improvement, it is the only one of which the fifth of Eu سلم clid will admit. In other respects I have followed Dr Sim- Book V. fon's edition, to the accuracy of which it would be difficult to make any addition. In one thing I must observe, that the doctrine of proportion, as laid down here, is meant to be more general than in Euclid's Elements. It is intended to include the properties of proportional numbers as well as of all magnitudes. Euclid has not this defign, for he has given a definition of proportional numbers in the seventh Book, very different from that of proportional magnitudes in the fifth; and it is not easy to justify the logic of this manner of proceeding; for we can never speak of two numbers and two magnitudes both having the same ratios, unless the word ratio have in both cafes the same signification. All the propositions about proportionals here given are therefore understood to be applicable to numbers; and accordingly, in the eighth Book, the proposition that proves equiangular parallelograms to be in a ratio compounded of the ratios of the numbers proportional to their fides, is demonstrated by help of the propositions of the fifth Book. On account of this, the word quantity, rather than magnicude, ought in strictness to have been used in the enunciation of these propofitions, because we employ the word quantity Co denote, not only things extended, to which alone we give the name of magnitudes, but also numbers. It will be suffiient, however, to remark, that all the propositions respecting the ratios of magnitudes relate equally to all things of which nultiples can be taken, that is, to all that is usually expressed by the word quantity in its most extended fignification, taking care always to observe, that ratio takes place only among ike quantities. (See Def. 4.) DEF. X. The definition of compound ratio was first given accuratey by Dr Simfon; for, though Euclid used the term, he did fo without defining it. I have placed this definition before those f duplicate and triplicate ratio, as it is in fact more general, nd as the relation of all the three definitions is best seen when Gc3 Book when they are ranged in this order, and expressed in the man. It was justly observed by Dr Simfon, that the expreffion, Book VI. TH BOOK VI. DEFINITION II. HIS definition is changed from that of reciprocal figures, PROP. XXVII. XXVIII. XXIX. As confiderable liberty has been taken with these propofi- It next occurred, that the problems themselves in the Book VI. 28th and 29th propositions are proposed in a more general form than is necessary in an elementary work, and that, therefore, to take those cases of them that are the most ufeful, as they happen to be the most simple, must be the best way of accommodating them to the capacity of a learner. The problem which Euclid proposes in the 28th is, "To a given straight line to apply a parallelogram equal to "a given rectilineal figure, and deficient by a parallelogram " fimilar to a given parallelogram;" which also might be more intelligibly enunciated thus: "To cut a given line, so " that the parallelogram that has in it a given angle, and that " is contained under one of the segments of the given line, " and a straight line which has a given ratio to the other seg ment, may be equal to a given space;" instead of which problem I have fubstituted this other; "to divide a given "straight line so that the rectangle under its segments may " be equal to a given space." In the actual solution of problems, the greater generality of the former proposition is an advantage more apparent than real, and is fully compenfated by the fimplicity of the latter, to which, also, it is always eafily reducible. The fame may be faid of the 29th, which Euclid enunciates thus: "To a given straight line to apply a parallelogram equal to a given rectilineal figure, exceeding by a pa"rallelogram similar to a given parallelogram." This might be proposed otherwise; "to produce a given line, so that the "parallelogram having in it a given angle, and contained by " the whole line produced, and a straight line that has a gi" ven ratio to the part produced, may be equal to a given " rectilineal figure." Instead of this, is given the following problem, more simple, and, as was observed in the former instance, very little less general: "To produce a given straight " line, so that the rectangle contained by the legments, be"tween the extremities of the given line, and the point to "which it is produced, may be equal to a given space." Book VI. 10: Book VII. Y PROP. A, B, C, &mo 4 There are eight propositions added to this Book, on account of their utility and their connection with this part of the elements. The first four of them are in Dr Simfon's edition, and among these Prop. A is given immediately after the third, being, in fact, a second cafe of that propofition, and capable of being included with it, in one enunciation. Prop. D is remarkable for being a theorem of Ptolemy the astronomer, in his Μεγαλη Συνταξις, and the foundation of the conftruction of his trigonometrical tables. Prop. E is the fimplest cafe of the former; it is also useful in trigonometry, and, under another form, was the 97th, or, in some editions, the 94th of Euclid's Data. The propofitions F and G are very useful properties of the circle, and are taken from the Loci Plani of Apollonius. Prop. H is a very remarkable proper ty of the triangle. : 2 ! BOOK VII. The reason for departing from Euclid in the geometry of folids has been already explained in the Preface, so that it only remains to make a few remarks on some particular definitions and theorems. DEF. VIII. and PROP. XX. Solid angles, which are defined here in the same manner as in Euclid, are magnitudes of a very peculiar kind, and may be remarked for not admitting of that accurate comparison, one with another, which is common in the other subjects of geometry. |