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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 another of his works to have availed himself of the idea of indefinitude with the fame fuccefs, viz. in his books of Porisms, which, have been reftored by Dr Simson, and in which the whole analysis turned on that idea, as I have thewn at length, in the third volume of the Transactions of the Royal Society of Edinburgh. The investigations of thofe propofitions were founded entirely on the principle of certain magnitudes admitting of innumerable values; and the methods of reafoning concerning them feem to have been extremely fimilar to thofe employed in the fifth of the Elenents. It is curious to remark this analogy between the dif ferent works of the fame author; and to confider, that the fkill, 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 fo well in treating the ftill more difficult fubject of Porisms.

With fuch an opinion of Euclid's manner of treating proportion, as I have now expreffed, it was impoffible that I fhould attempt to change any thing in the principle of his demonftrations. I have only fought to improve the language of them, by introducing a. concife mode of expreffion, of the fame nature with that which we ufe in arithmetic, and in algebra. Ordinary language conveys the ideas of the dif ferent operations fuppofed to be performed in thefe demonftrations fo flowly, and breaks them down into fo many parts, that they make not a fufficient impreffion on the understanding. This, indeed, will generally happen when the things treated of are not reprefented to the fenses by Diagrams, as they cannot be when we reafon 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 fhortnefs, and the rapidity with which it places objects before in the best manner pofmakes us, up fible for being merely a conventional language, and uling fymbols that have no refemblance to the things expreffed by them. Such a language, therefore, I have endeavoured to introduce here; and, I am convinced, that if it fhall be found an improvement, it is the only one of which the fifth of Eu

clid will admit. In other refpects 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 muft obferve, 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 he feventh 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 fame ratios, unless the word ratio have in both cafes the fame fignification. All the propofitions about proportionals here given are therefore understood to be applicable to numbers; and accordingly, in the eighth Book, the propofition that proves quiangular parallelograms to be in a ratio compounded of the ratios of the numbers proportional to their fides, is demonftrated by help of the propofitions of the fifth Book.

On account of this, the word quantity, rather than magniude, ought in ftrictness to have been used in the enunciation of these propofitions, because we employ the word quantity to denote, not only things extended, to which alone we give the name of magnitudes, but also numbers. It will be fufficient, however, to remark, that all the propofitions refpecting

ratios of magnitudes relate equally to all things of which multiples can be taken, that is, to all that is usually expreffed by the word quantity in its moft extended fignification, taking care always to obferve, that ratio takes place only among ike quantities. (See Def. 4.)

DE F. X.

The definition of compound ratio was first given accuratey by Dr Simfon; for, though Euclid used the term, he did so ithout 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

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Book V. when they are ranged in this order, and expreffed in the ma
ner done here; duplicate ratio being called a ratio compound-
ed of two equal ratios, and triplicate of three equal ratios, &c.

It was justly obferved by Dr Simfon, that the expreffio:
compound ratio is introduced merely to prevent circumlocution
and for the fake principally of enunciating those propofition
with concifenefs that are demonftrated by reasoning ex equo,
that is, by reasoning from the 22d or 23d of this Book. This
will be evident to any one who confiders carefully the Prop.E
of this, or the 23d of the 6th Book.

Book VI.

BOOK VI.

DEFINITION II.

HIS definition is changed from that of reciprocal figures,
which was of no ufe, to one that correfponds to the
language used in the 14th and 15th propofitions, and in other
parts of geometry.

PROP.

XXVII. XXVIII. XXIX.

As confiderable liberty has been taken with these propofi
tions, it is neceffary that the reasons for doing so should be ex
plained. In the first place, when the enunciations are tran
lated literally from the Greek, they found very harfhly, and
are, in fact, extremely obfcure. The phrafe of applying to
a straight line, a parallelogram deficient, or exceeding by ano
ther parallelogram, is fo elliptical and fo little analogous to
ordinary language, that there could be no doubt of the pro
priety of at least changing the enunciations.

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It next occurred, that the problems themselves in the Book VI. 28th and 29th propofitions are proposed in a more general form than is neceffary in an elementary work, and that, therefore, to take those cafes of them that are the most ufeful, as they happen to be the moft fimple, must be the best way of accommodating them to the capacity of a learner. The problem which Euclid propofes 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 alfo might be more intelligibly enunciated thus: "To cut a given line, fo "that the parallelogram that has in it a given angle, and that " is contained under one of the fegments 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;" inftead of which problem I have fubftituted this other; "to divide a given ftraight line fo that the rectangle under its fegments may "be equal to a given space." In the actual folution of problems, the greater generality of the former propofition 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.

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The fame may be faid of the 29th, which Euclid enunciates thus: "To a given ftraight line to apply a parallelogram equal to a given rectilineal figure, exceeding by a parallelogram fimilar to a given parallelogram." This might be propofed otherwife; "to produce a given line, fo that the parallelogram having in it a given angle, and contained by "the whole line produced, and a ftraight line that has a gi"ven ratio to the part produced, may be equal to a given "rectilineal figure." Inftead of this, is given the following problem, more fimple, and, as was obferved in the former inftance, very little lefs general: "To produce a given straight "line, fo that the rectangle contained by the fegments, between the extremities of the given line, and the point to "which it is produced, may be equal to a given space."

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Book VI.

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There are eight propofitions added to this Book, on ac count 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 thefe Prop. A is given immediately after the third, being, in fact, a fecond 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 aftronomer, in his Meyann Euvragis, and the foundation of the conftruction of his trigonometrical tables. Prop. E is the fimpleft cafe of the former; it is alfo ufeful in trigonometry, and, under another form, was the 97th, or, in fome editions, the 94th of Euclid's Data. The propofitions F and G are very ufeful properties of the circle, and are taken from the Loc Plani of Apollonius. Prop. H is a very remarkable property of the triangle.

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Book VII.

BOOK VII.

The reafon for departing from Euclid in the geometry of folids has been already explained in the Preface, fo that it only remains to make a few remarks on fome particular definitions and theorems.

DEF. VIII. and PRO P. XX.

Solid angles, which are defined here in the fame 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 fubjects of

geometry.

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