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No. 1761. July 27, 1861.

THE ATHENÆUM

The Formulæ of Plane Trigonometry, printed with Symbols (instead of Words) to express the “Goniometrical Ratios.” By C. W. Dodgson. (Oxford, J. H. & J. Parker.)

The Mathematical Lecturer of Christ Church, Oxford, has here published a little tract, which would have had a short notice in another part of our journal, but for a peculiarity which has a useful and praiseworthy character. Desiring to introduce certain changes in trigonometrical notation, he seeks the opinion of the mathematical world before he commits himself to a regular treatise. Now, first, the improvement of mathematical notation does not receive the discussion due to it; next, the plan adopted by Mr. Dodgson is one of the best which could be suggested. And it would be well worth the while of the proper Section of the British Association to make Notation in general the subject of conversation.

That nondescript animal the “general reader” is supposed to lump up sines and cosines with other things, in the manner of Swift:—

Sines, tangents, secants, radius, cosines, Subtangents, segments, and all those signs; Enough to prove that he who read 'em Was just as mad as he who made 'em.

But even the general reader may have almost arrived at the knowledge that sine, cosine, &c., are names given to the proportions which exist between the sides of a right-angled triangle.

The following are the names of the symbols, the usual abbreviations, and Mr. Dodgson's proposed symbols:—

Sine of &THgr; sin &THgr; [mathematical symbol] &THgr; Cosine of &THgr; cos &THgr; [mathematical symbol] &THgr; Tangent of &THgr; tan &THgr; [mathematical symbol] &THgr; Cotangent of &THgr; cot &THgr; [mathematical symbol] &THgr; Secant of &THgr; sec &THgr; [mathematical symbol] &THgr; Cosecant of &THgr; cosec &THgr; [mathematical symbol] &THgr; Versed sine of &THgr; vers &THgr; [mathematical symbol] &THgr;

On this suggestion, Mr. Dodgson asks three questions,—to two of which we shall subjoin our own answers:—1. Do you object in limine to the introduction of any symbols whatever as substitutes for the words sin, cos, &c.? We answer that we have no objection whatever to the suggestion; but that as to the introduction, we shall be guided by our opinion of the value of the proposed symbols. But by suggestion we mean such suggestion as Mr. Dodgson's. We very much disapprove of fundamental changes put forward in elementary works, especially when the proposer is a teacher who has the power of forcing his symbols into use in a particular college or school.

2. If not, can you suggest others better adapted than these for such a purpose? We think, and without any doubt, that we can. All the proposed symbols require the pen to be taken off in their formation: and we should utterly reject, as equally cumbrous with the ordinary abbreviations of current writing, any symbols which demand two acts of the pen. Now, in the current writing of sin, cos, tan, &c., the only nuisances are the dotting of is and the crossing of ts. We are sure that the cos of ordinary writing is preferable to Mr. Dodgson's substitute. We consider this of more importance than close attention to analogy in the formation of the symbols. But we think this need not altogether be lost sight of. At once, we can suggest what we should very much prefer to the system before us. For the sine, a circle with a dash downwards, like the p of Mayor's shorthand. For the cosine, a dash beginning with a hook, or pothook, as the writing-masters used to call it. For the tangent, a junction of these symbols, made of a circle, a dash and a pothook. For the cosecant, secant and cotangent, the preceding symbols inverted. For the versed sine, a dash, a horizontal stroke and the symbol of the cosine: a likeness to [mathematical symbol] — cos for a symbol which is very rarely used. All the symbols to be rather in larger style than the letters, in imitation of the integral sign, which was originally the long Italic S.

The third question asked has reference only to the supposition of some approval of the particular signs suggested.

We do not say that we could, on further consideration, suggest nothing better: nor do we say that we would at once countenance any change at all. We should require much discussion to make up our minds on either point.