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Viewing in this light Euclid's manner of treating propor- Book V. tion, I had no desire to change any thing in the principle of his demonstrations. I have only sought to improve the language of them, by introducing a concise mode of expression, 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 these demonstrations so slowly, and breaks them down into so many parts, that they make not a sufficient impression 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 possible for being merely a conventional language, and using symbols that have no resemblance to the things expressed 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 which the fifth Book of Euclid will admit. In other respects I have followed Dr. Simson's edition, to the accuracy of which it would be difficult to make any addition.
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 design, 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 cases the same sig, nification. All the propositions about proportionals here given are therefore understood to be applicable to numbers; and, accordingly, in the first book of the Supplement, the proposition that proves equiangular parallelograms to be in a ratio compounded of the ratios of the numbers proportional to their sides, is demonstrated by help of the propositions of the fifth book.
On account of this the word quantity, rather than mergnetude, ought in strictness to have been used in the enunciation
Book V. of these propositions, 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 sufficient, however, to remark, that all the propositions respecting the ratios of magnitudes relate equally to all things of which multiples can be taken, that is, to all which is usually expressed by the word quantity in its most extended signifia. cation. But it must always be remembered, that ratio takes place only among like quantities. (See Def. 4.)
The definition of compound ratio was first given accurately by Dr. Simson; for, though Euclid used the term, he did so without defining it. I have placed this definition before those of duplicate and triplicate ratio, as it is in fact more general, and as the relation of all the three definitions is best seen when they are ranged in this order. It is then plain that two equal ratios compound a ratio duplicate of either of them; three equal ratios, a ratio triplicate of either of them, &c.
It was justly observed by Dr. Simson that the expression, compound ratio, is introduced merely to prevent circumlocution, and for the sake principally of enunciating those propositions with conciseness that are demonstrated by reasoning, ex'æquo, that is, by reasoning from the 22d or 23d of this book. This will be evident to any one who considers carefully prop. F of this, or the 23d of the 6th book.
An objection which naturally ocćurs to the use of the term compound ratio, arises from its not being évident how the ratios described in the definition determine in any way the ratio which they are said to compound, since the magnitudes compounding them are assumed at pleasure. To remove this difficulty, it may be of use to state the matter thus : If there be any number of ratios (among magnitudes of the same kind) such that the consequent of
of them is the antecedent of that which immediately follows, the first of the antecedents has to the last of the consequents a ratio which evidently depends on the intermediate ratios, because if they be determined, it is determined also; and this dependence of one ratio on all the other ratios is expressed by-saying that it is
A B is evidently determined by the ratios &c. because if
B’ c each of the latter is fixed and invariable, the former cannot change. The exact nature of this dependence, and how one
is determined by the other, it is not the business of the definition to explain, but merely to give a name to a relation which it may be of importance to consider more atten tively.
THIS definition is changed from that of reciprocal Book VI. figures, which was of no use, to one that corresponds to the language used in the 14th and 15th propositions, and in other parts of geometry.
PROP. XXVII. XXVIII. XXIX.
As considerable liberty has been taken with these propositions, it is necessary that the reasons for doing so should be explained. In the first place, when the enunciations are translated literally from the Greek, they sound very harshly, and are, in fact, extremely obscure. The phrase of applying to a straight line a parallelogram deficient, or exceeding by
Book VI. another parallelograr., is so elliptical and so little analogous
to ordinary language, that there could be no doubt of the propriety of at least changing the enunciations.
It next occurred that the Problems themselves in the 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 which are the most useful, 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 īs straight line to apply a parallelogram equal to a given rec“ tilineal figure, and deficient by a parallelogram similar to a “ given parallelogram :” which also might be more intelligibly enunciated thus : “ To cut a given line, so that the paral“ lelogram 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 segment, may be “ equal to a given space;" instead of which problem I have substituted 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 compensated by the simplicity of the latter, to which it is always easily reducible.
The same may be said 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 parallelo
gram 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 “ given 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 segments between the extremities of the given line and the
point to which it is produced, may be equal to a given “ space.”
PROP. A, B, C, &c.
Ten propositions are added to this Book, on account of the utility and connection with this part of the Elements. The first four are in Dr. Simson's edition, and among these Prop. A is given immediately after the third, being, in fact, a second case of that proposition, 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 construction of his trigonometrical tables. Prop. E is the simplest case of the former, and is useful in trigonometry. Propositions F and G are very useful properties of the circle, and are taken from the Loci Plani of Apollonius. H and K are remarkable properties of the triangle. L is a proposition which, though it has been hitherto considered as belonging particularly to trigonometry, is so often of use in other parts of the mathematics, that it may be properly ranked among the elementary theorems of geometry.
PROP. V. and VI. &c.
THE demonstrations of the 5th and 6th propositions Book I. require the method of exhaustions, that is to say, they prove a certain property to belong to the circle, because it belongs to the rectilineal figures inscribed in it, or described about it according to a certain law, in the case when those figures approach to the circle so nearly as not to fall short of it, or to exceed it by any assignable difference. This principle is gene