The (almost really) Complete Works of Lewis Carroll

Introduction

In teaching the subject of Algebraical Geometry, I have found the advantage of giving formal definitions of such words as “axis,” “ordinate,” and “radius-vector,” as giving the learner a clearer notion of the elements of the subject than he could gather for himself from the ordinary treatises.

In drawing up these definitions I found that several of the geometrical terms borrowed from Euclid could not be used in Algebraical Geometry without some modification of their definitions as given by him, whilst other terms were wanted for which he had given no definition at all; hence I was led to prefix to my Definitions of Algebraical Geometry a modernised version of Euclid’s Definitions. The Propositions also of the subject appeared to me to be wanting in uniformity; for example, the relation between an equation of the first degree and a right line is discussed by one writer in the form “to find the equation to a straight line,” (thus assuming the possibility of such an equation being found;) by another, in the form “we proceed to enquire the geometrical signification of a single equation between the co-ordinates.” To remedy such defects as these I endeavoured to reduce the subject to a uniform series of formal Propositions. Such was the origin of the present Syllabus.

I hope it will not be thought presumptuous in me if I here state in order those defects which appear to me to exist in the modern treatises, and for which I have attempted to furnish a remedy; next, the method I have pursued in doing so; and lastly, the defects which still remain unremedied, and to which those, who are desirous of forwarding towards perfection this beautiful science, would do well to turn their attention.

The defects which I have attempted to remedy relate to three main points in the subject; I. Definitions, II. Enunciations, and III. Treatment of Subject.

I. Definitions. Many of the technical terms employed by the various writers on the subject are left undefined, or only with such definitions as may be gathered from a page or more of explanatory matter; others are inadequately defined, (as, for instance, the angle employed in Polar Co-ordinates, the definition of which, as given in Mr. Salmon’s Conic Sections, p. 9, will not apply to the case of a negative radius-vector); and even when a correct and formal definition is furnished, it is only to be found in the passage where the term first occurs, a method likely to give much unnecessary trouble in referring to it afterwards.

II. Enunciations. These, in the present treatises, are only given along with their deinonstrations, which again are interspersed with much explanatory and illustrative matter, all liable to confuse the student who desires to review the subject, briefly and connectedly, as a whole. Moreover there is a great want of uniformity in the language employed, the data being put first in some, and the quæsita in others; while some propositions have no “general enunciation” at all, but begin at once with the “particular enunciation.”

III. Treatment of Subject. The propositions are in too many cases made to depend on each other, so that if we wish to make out the proof of any particular theorem by itself, it is necessary to refer back to the proposition from which it is deduced, and from that perhaps to another, and so at last to build up the required proof. This method would not be so liable to objection if there were any received text-book on the subject, answering to Euclid in Geometry, and furnishing a recognised series of consecutive propositions which could be appealed to as authority; but, as the case now stands, when one writer may prove A from B, and another B from A, the whole system is illogical and therefore unsatisfactory.

Again, an unphilosophical method is employed in arriving at various of the necessary formulæ; it is a method neither analytical nor synthetical, but begins in synthesis, and concludes with a kind of analysis, introduced to prove the lawfulness of the assumption made, but giving no information as to how it was originally arrived at. As an instance, we may take the conversion of the equation $Ax+By+C=0$, into the form of $x.cosa+y.sina-\rho =0$, where the rule is laid down “divide by $\sqrt{A^2+B^2}$,” and we are afterwards told that we may assume $\frac{A}{\sqrt{A^2+B^2}}=cosa$, and $\frac{B}{\sqrt{A^2+B^2}}=sina$ “since the sum of the squares of these two quantities $=1$.” Again, the focus of the ellipse is introduced by arbitrarily taking two points on the major axis, at a distance from the centre equal to $\pm \sqrt{a^2-b^2}$, and discussing their properties; but no reason is given why these, rather than any other points, are so chosen for discussion.

A third deficiency is to be found in the notation adopted, where the symbols and language are not adapted to the expansion, which is necessary to meet the larger requirements of Algebraical Geometry of three dimensions. For instance, in the equations, $\frac{x-x^{\prime }}{l}=\frac{y-y^{\prime }}{m}$, and $y=tx+b$, when referred to rectangular Cartesian axes, it is usual to define “l” and “m” as “the cosine and sine of the angle which the line makes with the X-axis,”¹ and “t” as “the tangent of the same angle.” On coming to oblique axes, it is found necessary to modify both definitions; “l” and “m” become “the ratios which the sines of the angles, made by the line with the two axes respectively, bear to the sine of the angle between the axes,” and “t,” “the ratio between these two sines themselves.” On examination it appears that the definitions for rectangular axes are included as a particular case of these, and it may now be thought that we have got definitions sufficiently broad for the whole subject, but on coming to Solid Geometry we have to return to the cosines for rectangular axes, (as otherwise the phrase “the sine of the angle between the axes” would be unmeaning,) while for oblique axes we are forced upon the entirely new theory of projections, unless we choose to adopt the following definition for “l,” “the ratio which the sine of the angle made by the line with the intersection of two planes, (namely, the plane in which the line itself and the X-axis lie, and that in which the Y-axis and Z-axis lie,) bears to the sine of the angle made by the X-axis with the same intersection”! This instance will shew how necessary it is to consider the future requirements of Solid Geometry in dealing with Plane Geometry.

The method I have pursued in endeavouring to remedy these defects may be most conveniently discussed under the same three headings.

I. As to Definitions. In those borrowed from Euclid I have altered no word of his, except where it was necessary, either to bring the language into uniformity with the other definitions, or to meet the new requirements of Algebraical Geometry. In the rest, I have always taken his as the model, being careful at the same time to provide for the subsequent expansion necessary in Solid Geometry. And these definitions I have placed all together, at the commencement of each Book.

II. As to Enunciations. These I have given by themselves, without their proofs, to afford the student a convenient means of reviewing the subject, and testing his knowledge of it, by taking each enunciation in succession, and considering whether or no his unassisted memory can supply the demonstration. Where any difficulty is found in doing this, he is recommended to add a marginal reference to the place where the demonstration may be found, whether in the text-book which he employs on the subject, or in the Appendix to this Syllabus. I have endeavoured to reduce all these enunciations to one uniform shape, always placing the “data” before the “quæsita” or “demonstranda.” Lastly, I have endeavoured to make all the enunciations “general;” and I trust it will be found that they have thus gained in uniformity what they have, in some instances, lost in brevity and clearness.

III. As to Treatment of Subject. I have endeavoured, for reasons which I have already mentioned, to render each proposition, as far as possible, independent of others; and being thus released from the necessity of stringing them together in a consecutive series, I was able to adopt what seemed the more natural principle of arrangement, of beginning with those which involved the fewest and simplest conditions, and so going on to the more complicated. Whether I was right in attempting the further simplification of separating Problems from Theorems, I am now doubtful; it introduced several difficulties which would not have otherwise arisen; still, the great advantages of clearness and convenience of reference made the experiment a plausible one.

Next, I have introduced every new formula in a method as purely analytical as I found possible. For instance, in arriving at the equation to a right line, I first investigate the method of representing a point (the original purpose for which the system of co-ordinates is introduced); then, on finding that two equations are necessary for this, it naturally follows that we should consider the geometrical signification of one such equation taken by itself. Similarly, the rule already alluded to for reducing the equation $Ax+By+C=0$, to the form $x.cosa+y.sina-\rho =0$, is found at once by simply applying the test that two equations shall represent the same line. And similarly, the co-ordinates of the “focus” of an ellipse may be easily found by solving the Problem “to find a point whose distance, from any given point on the curve, shall be a simple function of the abscissa of that given point.”

Another innovation will be found in the method of treating what is called by Mr. Salmon “abridged notation.” It appeared to me that the symbols “α” and “β,” employed in that system, are in no other sense abridged forms of “$x.cosa+y.sina-p$ &c., than that in which “x” is an abridged form of “$\rho .cos\theta$,” (the relation between a Cartesian and a Polar System,) or of “$x^{\prime }.cos\theta -y^{\prime }.sin\theta$,” (the relation between two rectangular Cartesian Systems inclined at an angle θ). In other words, it appeared to me simpler, and I have certainly found it to be so in practice, to constitute a new System of reference, which I have called the “Distantial” System, and only to introduce the fact that $\alpha =x.cosa+y.sina-p$, as a formula of transformation from a Distantial to a Cartesian System.

The distinction introduced between an infinitely small quantity and absolute zero needs perhaps a word of apology, as it is not noticed in other treatises, and indeed belongs more properly to the subject of Differential Calculus. Still it is, at worst, superfluous, and those who think it merely fanciful will find that it can easily be omitted, as nothing in the book is made to depend upon it.

Apology is also due for the number of new words I have introduced; I have not done so in any case without an apparent necessity, to avoid the constant repetition of a cumbrous periphrasis. I will here enumerate the new words introduced, with reference to the pages where their definitions may be found. I have endeavoured to make them, as far as possible, suggest their meaning by their derivations:

divisional-angles, and divisional-ratio 18
direction-angles 21
abscissa-ratio, ordinate-ratio, and co-ordinate-ratio 22
sagittal-line, sagittal-angles, and sagit 22, 23
interceptive, sagittal, and directional equations 29
fixed-radius 30
vectorial-line 30
distantial system, lines of reference, vertex of reference, principal portion, and referent 36
interceptor 44
co-radical circles 96

It remains that I should enumerate those defects in the subject, for which I have not been able here to furnish a remedy.

In the first place, there is something unphilosophical in the very groundwork of the science. It professes to prove, from independent sources, many of the propositions of Pure Geometry, while it is so entirely dependent upon that science for its merest elements, that even the equation to a straight line cannot be investigated without assuming Euclid, Book VI, Prop. 4. As long as Algebraical Geometry requires the previous proof of so large a range of geometrical propositions, it cannot fairly be employed to prove any theorems which fall within that range.

Secondly, there are terms employed in the science in more senses than one; for the secondary meaning a new word should be introduced. For example, while we have “superficies” to denote a surface of indefinite extent, and “figure" to denote a limited portion of one—while we have “space” to denote indefinite extent of three dimensions, and “solid” to denote a limited portion of space—the word “line” is made to do double duty. Another example may be found in the word “direction,” discussed in the Appendix, p. 124.

Thirdly, there are several instances of two or more words being used in precisely the same sense; these require to be desynonymised, or, if that be impossible, all the synonymous terms, but one, should be excluded from use, and reserved for the possible future requisitions of the subject. As instances of this may be mentioned,

Fourthly, some simple rules are required for ascertaining, in any given problem, whether it may best be solved by the Cartesian, Polar, or Distantial System.

Many defects in the present Syllabus have been discovered and rectified while the sheets were passing through the press; many more, in an attempt so new, and (so to speak) experimental, have no doubt escaped my notice; there is also a great hiatus in the subject of Trigonometry, the whole of which needs to be systematised in the same way. Still I have thought it better to publish it in its present incomplete state, in order that by bringing it under the notice of more mathematicians than would otherwise have seen it, and so getting more suggestions for its improvement than could otherwise have been obtained, means may be afforded, whether to myself or to some other writer, of hereafter reducing the whole subject to a complete and uniform system, which shall occupy, with regard to Algebraical Geometry, the same position which is occupied by that of Euclid with regard to Pure Geometry.