The (almost really) Complete Works of Lewis Carroll

Part I. Addressed to the Teacher.

In preparing this edition of the first Two Books of Euclid, my aim has been to show what Euclid’s method really is in itself, when stripped of all accidental verbiage and repetition. With this object, I have held myself free to alter and abridge the language wherever it seemed desirable, so long as I made no real change in his methods of proof, or in his logical sequence.

This logical sequence, which has been for so many centuries familiar to students of Geometry—so that ‘The Forty-Seventh Proposition’ is as clear a reference as if one were to quote the enuntiation in full—it has lately been proposed to supersede: partly from the instinctive passion for novelty which, even if Euclid’s system were the best possible, would still desire a change; partly from the tacitly assumed theory that modern lights are necessarily better than ancient ones. I am not now speaking of writers who retain unaltered Euclid’s sequence and numbering of propositions, and merely substitute new proofs, or interpolate new deductions, but of those who reject his system altogether, and, taking up the subject de novo, attempt to teach Geometry by methods of their own.

Some of these rival systems I have examined with much care (I may specify Chauvenet, Cooley, Cuthbertson, Henrici, Legendre, Loomis, Morell, Pierce, Reynolds, Willock, Wilson, Wright, and the Syllabus put forth by the Association for the Improvement of Geometrical Teaching), and I feel deeply convinced that, for purposes of teaching, no treatise has yet appeared worthy to supersede that of Euclid.

It can never be too constantly, or too distinctly, stated that, for the purpose of teaching beginners the subject-matter of Euclid I, II, we do not need a complete collection of all known propositions (probably some thousands) which come within that limit, but simply a selection of some of the best of them, in a logically arranged sequence. In both these respects, I hold that Euclid’s treatise is, at present, not only unequalled, but unapproached.

For the diagrams used in this book I am indebted to the great kindness of Mr. Todhunter, who has most generously allowed me to make use of the series prepared for his own edition of Euclid.

I will here enumerate, under the three headings of ‘Additions,’ ‘Omissions’ and ‘Alterations,’ the chief points of difference between this and the ordinary editions of Euclid, and will state my reasons for adopting them.

1. Additions.

Def. &c. § 11. The Axiom ‘Two different right Lines cannot have a common segment’ (in 3 equivalent forms). This is tacitly assumed by Euclid, all through the two Books (see Note to Prop. 4), and it is so distinctly analogous to his ‘two right Lines cannot enclose a Superficies’ that it seems desirable to have it formally stated.

Def. &c. § 20. Here, to Euclid’s Postulate ‘A Circle can be described about any Centre, and at any distance from it,’ I have added the words ‘i. e. so that its Circumference shall pass through any given Point.’ This I believe to be Euclid’s real meaning. Modern critics have attempted to identify this given ‘distance’ with ‘length of a given right Line,’ and have then very plausibly pointed to Props. 2, 3, as an instance of unnecessary length of argument. ‘Why does he not,’ they say, ‘solve Prop. 3 by simply drawing a Circle with radius equal to the given Line?’ All this involves the tacit assumption that the ‘distance’ (διάστημα, i. e. ‘interval’ or ‘difference of position’) between two Points is equal to the length of the right Line joining them. Now it may be granted that this ‘distance’ is merely an abbreviation for the phrase ‘length of the shortest path by which a Point can pass from one position to the other:’ and also that this path is (as any path would be) a Line: but that it is a right Line is just what Euclid did not mean to assume: for this would make Prop. 20 an Axiom. Euclid contemplates the ‘distance’ between two Points as a magnitude that exists quite independently of any Line being drawn to join them (in Prop. 12 he talks of the ‘distance $CD$’ without joining the points C, D), and, as he has no means of measuring this distance, so neither has he any means of transferring it, as the critics would suggest. Hence Props. 2, 3, are logically necessary to prove the possibility, with the given Postulates, of cutting off a Line equal to a given Line. When once this has been proved, it can be done practically in any way that is most convenient.

Axioms, § 9. This is quite as axiomatic as the one tacitly assumed by Euclid (in Props. 7, 18, 21, 24), viz. ‘If one magnitude be greater than a second, and the second greater than a third: the first is greater than the third.’ Mine is shorter, and has also the advantage of saving a step in the argument: e. g. in Prop. 7, Euclid proves that the angle $ADC$ is greater than the angle $BCD$, a fact that is of no use in itself, and is only needed as a step to another fact: this step I dispense with.

Prop. 8. Here I assert of all three angles what Euclid asserts of one only. But his Proposition virtually contains mine, as it may be proved three times over, with different sets of bases.

Prop. 24. Euclid contents himself with proving the first case, no doubt assuming that the reader can prove the rest for himself. The ordinary way of making the argument complete, viz. to interpolate ‘of the two sides $DE$, $DF$, let $DE$ be not greater than $DF$,’ is very unsatisfactory: for, though it is true that, on this hypothesis, F will fall outside the Triangle $DEG$, yet no proof of this is given. The Theorem, as here completed, is distinctly analogous to Prop. 7.

Book II, Def. § 4. The introduction of this one word ‘projection’ enables us to give, in Props. 12, 13, alternative enuntiations which will, I think, be found much more easy to grasp than the existing ones.

Book II, Prop. 8. Considering that this Proposition, with the ordinary proof, is now constantly omitted by Students, under the belief that Examiners never set it, I venture to suggest this shorter method of proving it, in hopes of recalling attention to a Theorem which, though not quoted in the Six Books of Euclid, is useful in Conic Sections.

[Instead of ‘On $AD$ describe’ &c, read as follows:—

Square of $AD$ is equal to
squares of $AB$, $BD$, with twice rectangle of $AB$, $BD$; [II. 4.
i. e. to squares of $AB$, $BC$, with twice rectangle of $AB$, $BC$;
i. e. to twice rectangle of $AB$, $BC$, with square of $AC$, [II. 7.
with twice rectangle of $AB$, $BC$;
i. e. to four times rectangle of $AB$, $BC$, with square of $AC$. Q. E. D.]

2. Omissions.

Euclid gives separate Definitions for ‘plane angle’ and ‘plane rectilineal angle.’ I have ignored the existence of any angles other than rectilineal, as I see no reason for mentioning them in a book meant for beginners.

Prop. 11. Here I omit the Corollary (introduced by Simson) ‘Two Lines cannot have a common segment,’ for several reasons. First, it is not Euclid’s: secondly, it is assumed as an Axiom, at least as early as Prop. 4: thirdly, the proof, offered for it, is illogical, since, in order to draw a Line from B at right angles to $AB$, we must produce $AB$; and as this can, ex hypothesi, be done in two different ways, we shall have two constructions, and therefore two perpendiculars to deal with.

Prop. 46. Here instead of drawing a Line, at right angles to $AB$, longer than it, and then cutting off a piece equal to it, I have combined the two processes into one, following the example which Euclid himself has set in Prop. 16.

3. Alterations.

Definitions, &c. § 7. Instead of the usual ‘A straight Line is that which lies evenly between its extreme Points,’ I have expressed it ‘A right Line,’ (‘right’ is more in harmony, than ‘straight,’ with the term ‘rectilineal’) ‘is one that lies evenly as to Points in it.’ This is Euclid’s expression: and it is applicable (which the other is not) to infinite Lines.

Prop. 12. Here I bisect the angle $FCG$ instead of the Line $FG$: i. e. I use Prop. 9 instead of Prop. 10. The usual construction really uses both, for Prop. 10 requires Prop. 9.

Prop. 16. Here, instead of saying that it ‘may be proved, by bisecting $BC$ &c. that the angle $BCG$ is greater than the angle $ABC$,’ I simply point out that it has been proved—on the principle that, when a Theorem has once been proved for one case, it may be taken as proved for all similar cases.

Prop. 30. Here Euclid has contented himself, as he often does, with proving one case only. But unfortunately the one he has chosen is the one that least needs proof: for, if it be given that neither of the outside Lines cuts the (infinitely producible) middle Line, it is obvious that they cannot meet each other.

Book II, Prop. 14. Here, instead of producing $DE$ to H, I have drawn $EH$ at right angles to $BF$. This at once supplies us with the fact that $GEH$ is a right angle, without the necessity of tacitly assuming, as Euclid does, that ‘if one of the two adjacent angles, which one Line makes with another, be right, so also is the other.’

Part II. Addressed to the Student.

The student is recommended to read the Two Books in the following order, making sure that he has thoroughly mastered each Section before beginning the next.