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to show that their powers of improvement were not exhausted.

While we uphold the opinion that it is indispensably necessary to treat the doctrine of proportion geometrically, and so highly esteem Euclid's method, let it not be supposed that we condemn the arithmetical or algebraical system of treatment, for we are convinced that the doctrine of proportion should be fully and fairly developed in every complete elements of arithmetic, of algebra, and of geometry; because it is a subject which belongs to one as much as to either of the other two; and on these three elementary sciences a whole course of mathematics is founded, through the entire of which the doctrine of proportion is mingled, and it matters not whether the structure be raised upon numbers, symbols, or lines, &c., the well-being of the combination must wholly depend on the capacity and firmness of the foundations.

For if we question the propriety of letting x, y, z, &c., stand for magnitudes, regardless whether they be incommensurable or not, we must question the whole of our beautiful system of analysis, which is just as certain in its results as plane geometry, and much more extensive in its application.

Nor do we defend the system of showing how inadequate numbers or letters are to express magnitudes, or their relations to one another; for it is admirable to see how the parallel propositions agree, although managed by means essentially different. To thoroughly

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understand the doctrine of proportion, it should first be acquired arithmetically, then algebraically, and both these methods made subservient to the right understanding the developement of geometrical proportion.

The introduction of symbols into works on geometry is every day becoming more general; and as by their assistance the demonstrations can be more perspicuously arranged, and the train of arguments exhibited more systematic and concise, it would therefore be unnecessary to offer any remark on their adoption in the present performance; besides, symbols, while recording each stage of the proposition faithfully, relieve the mind to contemplate the absolute quantities. But the symbols used in geometry must be considered not only as appropriate emblems of the quantities themselves, but also as expressive; and not as any measures or numerical values of them.

However, lest the student should confound these visible symbols with the abstractions for which they stand, let us take A x B in a geometrical sense: now we have no idea of the product of two numbers, but of a real rectangular space, comprehended under two right lines, represented by A and B, with two others equal to them, to complete the parallelogram.

B x C Nor is to be understood in the light of an algebraic fraction, but as a right line which is a fourth proportional to three other right lines, which are represented by A, B, and C.

And when we say, 0:0:0:0

O 0:0, we do not annex the idea to those,

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that the circle is to the square as the rhombus is to the sector; no, these quantities may represent any magnitude whatever, whose antecedents and consequents are homogeneous abstractedly considered: but in pure geometry regard is always had to lines, surfaces, or solids.

With respect to the explanation of signs used in this work, they need but little, as they are all in common use in algebraic notation; however, to be more particular, let us take a line from the demonstration of Prop. VII, viz. :—

“.. if M○=,=, or¬m then M, or ¬ m☐.” This in ordinary language would be thus expressed therefore if M times the magnitude represented by be greater, equal, or less than m times the magnitude represented by, then will M times the magnitude represented by◇ be greater, equal, or less than m times the magnitude represented by When these distinctions are understood, the method of expressing demonstrations symbolically will be equally logical, strict, and convincing, without being attended with that tediousness and circuitous detail which frequently accompany other methods.

Should the symbols which represent each magnitude be differently coloured, it would greatly aid the student in the demonstrations. This system was originally intended, but abandoned, on account of the great expense of printing in colours: the want may be easily supplied by the learner.

One particular more may be worthy of remark, that is, with respect to the fifth definition, which has ever been a stumbling-block to students commencing the fifth book, and a source of much controversy and dispute among mathematicians, both in ancient and modern times. In this however we have adopted some slight modification in the words of the original text, but not the slightest change in the nature of the definition; the alteration principally consists in the adoption of “every equimultiple,” instead of any equimultiple whatever;" for most beginners form a notion, from the last sentence, that any promiscuously-chosen equimultiple whatever would be sufficient to test whether four magnitudes were proportional or not; and seldom conceives that the conditions of the definition require to be fulfilled with every set of equimultiples that might be selected. This test for proportionals has been regarded by some as a paradox, or as a thing impossible to be applied, and by others quite foreign to the purpose; but it does not follow from this definition that an infinite number of trials must be made every time we want to test four proportionals; we only want to establish this as a standard, and when once allowed the difficulty is at once removed. This we presume will be fully established and readily comprehended from our enunciation of this definition.

Innumerable have been the attempts to elude or surmount the obstacle contained in this definition, but not one of those have been more successful than an

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other, and on a mature consideration it is evident that no other definition essentially different could have been given equally applicable and general; and it is very probable that even those definitions, which since the time of Euclid have been proposed as substitutes for the fifth, presented themselves to him, and that from their want of generality he was obliged to reject them. It is true, definitions should require no explanation, nor should they contain words which need themselves to be defined —they should be clear and perspicuous: notwithstanding this we have explained many of the definitions more familiarly, lest the most ordinary capacity should fail to comprehend them. To teach should be the highest aim of a writer on elementary subjects, and not to adopt (which is too often the case) that stiff and formal manner so prejudicial to and inconsistent with the ideas of a learner; every thing likely to embarrass should be explained, and that authorial kind of scientific dignity should be set aside when the object is to instruct others.

The following are the objects for which this work is published:--to uphold Euclid's fifth book as the only legitimate doctrine of geometrical proportion as yet produced ; to show that proportion should be treated of algebraically and arithmetically as well as geometrically, as it equally belongs to all; and to endeavour to clear, without destroying the universality and rigour of its conclusions, this extensive mathematical branch of that difficult, elaborate, and intricate reasoning

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