Proved Algrebraically so far as it relates to Commensurable Magnitudes to which is prefixed a Summary of all the necessary algebraical operations.
Preface
The student is recommended to go through this treatise in the following order:—
First, to master the ‘Preliminary Algebra,’ and not to go further until he finds that, when covering up the right-hand column and setting himself any question in the left-hand column, he can at once work out (not merely supply from memory) the required answer.
Secondly, taking the Algebraical Enimciation which stands at the top of the right-hand column in each Proposition, to learn to supply the proof which follows it. To do this, he should cover the rest of the right-hand column, and try to work out the proof for himself, with the help of the directions in the left-hand column. As every step of the work has been already done in the ‘Preliminary Algebra,’ this ought to be possible without any reference to the right-hand column: but if any difficulty should occur, there will usually be found a marginal reference to the ‘Preliminary Algebra,’ and it will be better to turn back to the section referred to, and so refresh the memory, than to look at the right-hand column, which should only be uncovered, when the proof has been written out, as a test of the correctness of the work.
Thirdly, to practise himself in working out the same proofs, without the help of the directions in the left-hand column, from the Algebraical Enunciations only. These are given by themselves at p. 49.
Fourthly, taking the Enunciation printed in small type in each Proposition, and covering up all below it, to learn to express it algebraically, as given in the first sentence of the right-hand column.
Fifthly, taking the Enunciation printed in large type in each Proposition, and covering up all below it, to learn to repeat it with the addition of algebraical symbols for the magnitudes, as given in the small-type Enunciation.
Sixthly, to learn Euclid’s Definitions and Axioms, given at p. 53.
Appendix
Euclid’s Definition of Proportion is not used in the Fifth Book, when proved algebraically, since this method of Proof applies to commensurable Magnitudes only, for which a much simpler Definition is sufficient: but it is required for Prop. I of the Sixth Book, on which the rest of that Book depends, so that some explanation of it may fitly be given here.
The Student should pay particular attention to the word “whatsoever.” Four magnitudes are said to be Proportionals, not merely when “any equimultiples” fulfil certain conditions (in which case one successful instance would be enough to justify the use of the name, in spite of many unsuccessful instances being found), but when “any whatsoever” fulfil them (in which case all instances must be successful to justify the use of the name, and a single unsuccessful instance would be enough to destroy our right to use it). To take an illustration from Chemistry, a drug might fairly be called “dangerous to life,” if any instance could be found of a person having died from swallowing it, but it could not be called “certainly fatal to life,” unless it could be shown that any person whatsoever, who swallowed it, must die in consequence; and a single proved case of a person having survived it would destroy our right to use the name.
In other words, before we have a right to call certain Magnitudes “Proportionals” in Euclid’s sense of the word, we must first have proved what is called in Logic “a Universal Proposition” (whose typical form is “All A are B”); that is, we must have proved that all equimultiples of these Magnitudes fulfil certain conditions.
Now a Universal Proposition may be proved by two totally distinct methods. One may be defined as “the enumeration of all instances,” the other as “the establishment of a general law.” Before explaining these phrases as applied to Euclid’s Definition, let us illustrate them by examples from another subject.
Suppose we wish to establish the Universal Proposition that “All English Queens who have reigned since the Conquest have had the letter ‘A’ in their names.” This may be proved true by enumerating all the names, and pointing out that each fulfils the condition stated. But, inasmuch as the circumstance is merely an accident as to each name, no “general law” can be shown to exist. In this case, then, we are restricted to the first method of proof.
Next, suppose the Proposition to be “All English Queens who have reigned since the Conquest have been of royal descent.” In this case we have a choice of methods: we may either enumerate all the names, giving the genealogy of each: or we may show that, by the principles of our Constitution, such royal descent is essential to a Queen. In many cases of this kind, it would be quite a matter for consideration, which method to employ: sometimes the one would be found the more convenient, sometimes the other.
Thirdly, suppose the Proposition to be “All English Queens, who have reigned since the Conquest, or who ever will reign, have had, or will have, weight.” In this case, since some of the instances referred to do not yet exist, so that no evidence, concerning them individually, can be given, we are restricted to the use of the second method: that is, we can only prove the Proposition by showing that all English Queens, past, present, and future, are necessarily human beings; and that human beings, by a law of Nature, have weight.
Now which of these two methods does Euclid mean us to employ, when he tells us, before we can use the name “Proportionals” of certain Magnitudes, to prove the Universal Proposition that “all equimultiples” fulfil certain conditions?
The number of equimultiples we may take of the magnitudes is infinite: hence “the enumeration of all instances” is impossible in this case, and the only method left us is “the establishment of a general law.”
To take the actual instance in which the Definition is used by Euclid—Prop. I of the Sixth Book. Euclid wishes to show that if two Triangles be of the same altitude, the two bases and the two Triangles constitute four “Proportionals.” To do this, out of the infinite number of possible equimultiples of the first and third, he chooses a single instance; and out of the infinite number of possible equimultiples of the second and fourth, he also chooses a single instance. He does not assume it to be enough for his purpose to show that, in this particular pair of instances, the test of “if greater, greater; if equal, equal; if less, less” is fulfilled. The essence of his argument consists in showing that what is true in the particular instances chosen would also, from the nature of the case, be true in every other conceivable instance which could be taken.
This part of the proof depends on two theorems—first, Prop. XXXYII of the First Book, namely “Triangles on equal bases, and between the same parallels, are equal”; secondly, on an easy deduction from this (which is not explicitly stated by Euclid, and so is often overlooked), namely, “Triangles between the same parallels, but on unequal bases, are unequal, that which is on the greater base being the greater.” Both Prop. XXXVII and this deduction from it are clearly of universal applicability. Hence Euclid’s proof of the Universal Proposition, that “all equimultiples” fulfil the requisite conditions, is complete, and his conclusion, that the original four Magnitudes are “Proportionals,” is legitimate.