APPENDIX.

Algebraic Method of demonstrating the Propositions in the fifth book of Euclid's Elements, according to the text and arrangement in Simson's edition.

SIMSON'S Euclid is undoubtedly a work of great merit, and is in very general use among mathematicians; but notwithstanding all the efforts of that able commentator, the fifth book still presents great difficulties to learners, and is in general less understood than any other part of the elements of Geometry. The present essay is intended to remove these difficulties, and consequently to enable learners to understand in a sufficient degree the doctrine of proportion, previously to their entering on the sixth book of Euclid, in which that doctrine is indispensable.

I have omitted the demonstrations of several propositions, which are used by Euclid merely as lemmata, but are of no consequence in the present method of demonstration.

Instead of Euclid's definition of proportion, as given in his 5th definition of the 5th book, I make use of the common algebraic definition; but I have shown the perfect equivalence of these two definitions. This perfect reciprocity between the two definitions is a matter of great importance in the doctrine of proportion, and has not (as far as I can learn) been discussed by any preceding mathematician.

With respect to compound ratio, I have also given another definition of it instead of that given by Dr. Simson; as his definition is found exceedingly obscure by beginners, and is in my judgment one of the most objectionable things in his edition of Euclid's Elements.

The literal operations made use of in the present paper are extremely simple, and require very little previous knowledge of algebra to render them intelligible.

The algebraic signs commonly used to indicate greater, equal, less, are 7,, : thus the three expressions ab, c=d, ef, signify that a is greater than b, that c is equal to d, and that e is less than f. The expression cd is called an equation or equality; the others ab, eƒ, are called ine qualities.

Also when four quantities are proportionals, we shall express this relation in the usual mode by points; thus,

A: BC: D

is to be read, A is to B as C is to D; or, A has the same ratio to B that C has to D.

THE ELEMENTS OF EUCLID, BOOK V.

Definitions.
I.

A less magnitude is said to be a part of a greater, when the less measures the greater, that is, when the less is contained a certain number of times exactly in the greater.

II.

A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly.

III.

Ratio is a mutual relation of two magnitudes of the same kind to one another in respect to quantity.

IV.

Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.

V.

The ratio of the magnitude A to the magnitude B is the number showing how often A contains B; or, which is the same thing, it is the quotient when A is numerically divided by B, whether this quotient be integral, fractional, or surd.

Explication.

This fifth definition, with its corollaries, is used in the present essay instead of Euclid's 5th and 7th definitions: the following examples will sufficiently illustrate the definition. Let A=20, and B≈5, then the ratio of A to B, or of 20 to 5, A 20

is or or 4, so that the ratio of 20 to 5 is 4. Again, let B

5

A=5, and B20, then

1

A

B

5 to 20 is. Lastly, let A=12/2, and B=4, then

12√2

4

*=3√2, and therefore the ratio of 12/2 to 4 is 3√2. COROLLARY I. If four magnitudes A, B, C, D, be so reA C it is evident the ratio of A to B is the same

lated that

BD'

with the ratio of C to D.

COR. II. Any four magnitudes whatever, so related that the ratio of the first to the second is the same with the ratio of the third to the fourth, may be expressed by

rA, A, rB, B ;

and

the first of the four being rA, the secoud A, the third rB, the fourth B; the magnitudes A and B being any whatever, and the letter r denoting each of the two equal ratios or quotients when the first rA is divided by the second A, and the third r B divided by the fourth B.

COR. III. When four magnitudes A, B, C, D, are so relat

A

C

ed that is greater than it is evident that the ratio of A

B

D

to B is greater than the ratio of C to D; or that the ratio of C to D is less than the ratio of A to B.

The Fifth Definition according to Euclid.

The first of four magnitudes is said to have the same ratio to the second which the third has to the four h, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth, if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.

SCHOLIUM. We shall demonstrate towards the close of this essay, that this deunition of Euclid's and our 5th definition, according to the common algebraic method, are not only consistent with each other, but also perfectly equivalent, each comprehending whatsoever is comprehended by the other.

VI.

When four magnitudes are proportionals, it is usually expressed by saying, the first is to the second as the third to the fourth.

The Seventh Definition according to Euclid.

When of the equimultiples of four magnitudes, (taken as in the fifth definition) the multiple of the first is greater than that of the second, but the multiple of the third is not greater than that of the fourth; then the first is said to have to the second a greater ratio than the third has to the fourth; and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second.

VIII.

Analogy or proportion is the equality of ratios.

IX.

Omitted.

X.

When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.

XI.

When four magnitudes are continued proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c. increasing the denomination still by unity in any number of propor

tionals.

Definition A, viz. of compound ratio, omitted.

XII.

In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another. XIII.

Permutando, or Alternando, by permutation, or by alternation, or alternately, are terms used, when of four proportionals it is inferred that the first is to the third as the second to the fourth.

XIV.

Invertendo, by inversion, or inversely, when of four proportionals, it is inferred that the second is to the first as the fourth to the third.

XV.

Componendo, by composition, when it is inferred that the sum of the first and second is to the second as the sum of the third and fourth is to the fourth.

XVI.

Dividendo, by division, when it is inferred that the excess of the first above the second is to the second as the excess of the third above the fourth is to the fourth.

XVII.

Convertendo, by conversion, or conversely, when it is inferred that the first is to its excess above the second, as the third to its excess above the fourth.

XVIII.

Ex æquali (sc. distantia), or ex æquo, from equality of distance, when there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred that the first is to the last of the first rank of magnitudes as the first is to

the last of the others of this there are the two following kinds, which arise from the different order in which the magnitudes are taken two and two.

XIX.

Ex æquali, from equality; this term is used simply by itself, when the first magnitude is to the second of the first rank, as the first to the second of the other rank, and the second to the third of the first rank as the second to the third of the other; and so on in order; and it is inferred that the first is to the last of the first rank as the first is to the last of the other rank.

XX.

Ex æquali, in proportione perturbata, seu inordinata, from equality in perturbate proportion this term is used when the first is to the second of the first rank as the last but one to the last of the other rank, and the second is to the third of the first rank as the last but two to the last but one of the other rank, and so on in a cross order; and it is inferred that the first is to the last of the first rank as the first is to the last of the other rank.

XXI.

If A, B, C, D, be any number of magnitudes of the same kind, and P any other magnitude; and if we make A : B :: PQ; and B C :: Q: R; and C :D :: RS; the ratio of P to S is said to be compounded of the ratios of A to B, B to C, C ́to D.

:

AXIOMS.

I. Equimultiples of the same, or of equal magnitudes, are equal.

II. These magnitudes of which the same, or equal magnitudes, are equimultiples, are equal to one another.

III. A multiple of a greater magnitude is greater than the same multiple of a less.

IV. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

PROPOSITIONS.

Propositions I. II. III. V. and VI. are omitted, as they do not treat of proportion, and are not wanted in the method of demonstration adopted in this essay.

PROP. IV. THEOR.

If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth; that is, the