APPENDIX

On Archimedes’s proof of the subtangent-property of a spiral.

THE section of the treatise On Spirals from Prop. 3 to Prop. 20 is an elaborate series of propositions leading up to the proof of the fundamental property of the subtangent corresponding to the tangent at any point on any turn of the spiral. Libri, doubtless with this series of propositions in mind, remarks (Histoire des sciences mathématiques en Italie, i, p. 31) that ‘Après vingt siècles de travaux et découvertes, les intelligences les plus puissantes viennent encore échouer contre la synthèse difficile du Traité des Spirales d’Archimède.’ There is nofoundation for this statement, which seems to be a too hasty generalization from a dictum, apparently of Fontenelle, in the Histoire de l’ Académie des Sciences pour l’année 1704 (p. 42 of the edition of 1722), who says ofthe proofs of Archimedes in the work On Spirals: ‘Elies sont si longues, et si difficiles à embrasser, que, comme on l’a pû voir dans la Préface de l’Analyse des Infiniment petits, M. Bouillaud a avoué qu’il ne les avoit jamais bien entendues, et que Viète les a injustement soupçonnées de paralogisme, parce qu’il n’avoit pû non plus parvenir à les bien entendre. Mais toutes les preuves qu’on peut donner de leur difficulté et deleur obscurité tournent à la gloire d’Archimède; car quelle vigueur d’esprit, quelle quantité de vûes différentes, quelle opiniâtreté de travail n’a-t-il pas fallu pour lier et pour disposer un raisonnement que quelques-uns de nos plus grands géomètres ne peuvent suivre, tout lié et tout disposé qu’il est?’

P. Tannery has observed¹ that, as a matter of fact, no mathematicians of real authority who have applied or extended Archimedes’s methods (such men as Huygens, Pascal, Roberval and Fermat, who alone could have expressed an opinion worth having), have ever complained of the ‘obscurity’ of Archimedes; while, as regards Vieta, he has shown that the statement quoted is based on an entire misapprehension, and that, so far from suspecting a fallacy in Archimedes’s proofs, Vieta made a special study of the treatise On Spirals and had the greatest admiration for that work.

But, as in many cases in Greek geometry where the analysis is omitted or even (as Wallis was tempted to suppose) of set purpose hidden, the reading of the completed synthetical proof leaves a certain impression of mystery; for there is nothing in it to show why Archimedes should have taken precisely this line of argument, or how he evolved it. It is a fact that, as Pappus said, the subtangent-property can be established by purely ‘plane’ methods, without recourse to a ‘solid’ νєσις (whether actually solved or merely assumed capable of being solved). If, then, Archimedes chose the more difficult method which we actually find him employing, it is scarcely possible to assign any reason except his definite predilection for the form of proof by reductio ad absurdum based ultimately on his famous ‘Lemma’ or Axiom.

It seems worth while to re-examine the whole question of the discovery and proof of the property, and to see how Archimedes’s argument compares with an easier ‘plane’ proof suggested by the figures of some of the very propositions proved by Archimedes in the treatise.

In the first place, we may be sure that the property was not discovered by the steps leading to the proof as it stands. I cannot but think that Archimedes divined the result by an argument corresponding to our use of the differential calculus for determining tangents. He must have considered the instantaneous direction of the motion of the point P describing the spiral, using for this purpose the parallelogram of velocities. The motion of P is compounded of two motions, one along OP and the other at right angles to it. Comparing the distances traversed in an instant of time in the two directions, we see that, corresponding to a small increase in the radius vector r, we have a small distance traversed perpendicularly to it, a tiny arc of a circle of radius r subtended by the angle representing the simultaneous small increase of the angle θ (AOP). Now r has a constant ratio to θ which we call a (when θ is the circular measure of the angle θ). Consequently the small increases of r and θ are in that same ratio. Therefore what we call the tangent of the angle OPT is r / a, i.e. OT / r = r / a; and OT = r² / a or rθ, that is, the arc of a circle of radius r subtended by the angle θ.

To prove this result Archimedes would doubtless begin by an analysis of the following sort. Having drawn OT perpendicular to OP and of length equal to the arc ASP, he had to prove that the straight line joining P to T is the tangent at P. He would evidently take the line of trying to show that, if any radius vector to the spiral is drawn, as OQ′, on either side of OP, Q′ is always on the side of TP towards O, or, if OQ′ meets TP in F, OQ′ is always less than OF. Suppose that in the above figure OR′ is any radius vector between OP and OS on the ‘backward’ side of OP, and that OR′ meets the circle with radius OP in R, the tangent to it at P in G, the spiral in R′, and TP in F′. We have to prove that R, R′ lie on opposite sides of F′, i.e. that RR′ > RF′; and again, supposing that any radius vector OQ′ on the ‘forward’ side of OP meets the circle with radius OP in Q, the spiral in Q′ and TP produced in F, we have to prove that QQ′ < QF.

Archimedes then had to prove that

(1)

(2)

Now (1) is equivalent to

But

therefore it was necessary to prove, alternando, that

(3)

Similarly, in order to satisfy (2), it was necessary to prove that

(4)

Now, as a matter of fact, (3) is a fortiori satisfied if

but in the case of (4) we cannot substitute the chord PQ for the arc PQ, and we have to substitute PG′, where G′ is the point in which the tangent at P to the circle meets OQ produced; for of course PG′ > (arc PQ), so that (4) is a fortiori satisfied if

It is remarkable that Archimedes uses for his proof of the two cases Prop.8 and Prop.7 respectively, and makes no use of Props 6 and 9, whereas the above argument points precisely to the use of the figures of the two latter propositions only.

For in the figure of Prop: 6 (Fig. 1), if OFP is any radius cutting AB in F and if PB produced cuts OT, the parallel to AB through O, in H, it is obvious, by parallels, that

Also PH becomes greater the farther P moves from B towards A, so that the ratio PF : PB diminishes continually, while it is always less than OB : BT (where BT is the tangent at B and meets OH in T), i.e. always less than BM : MO.

Hence the relation (3) is always satisfied for any point R′ of the spiral on the ‘backward’ side of P.

But (3) is equivalent to (1), from which it follows that F′R is always less than RR′, so that R′ always lies on the side of TP towards O.

Next, for the point Q′ on the ‘forward’ side of the spiral from P, suppose that in the figure of Prop. 9 or Prop. 7 (Fig. 2) any radius OP of the circle meets AB produced in F, and the tangent at B in G; and draw BPH, BGT meeting OT, the parallel through O to AB, in H, T.

Then

and obviously, as P moves away from B towards OT, i.e. as G moves away from B along BT, the ratio OG : GT increases continually, while, as shown, PF : BG is always > BM : MO and, a fortiori,

That is, (4) is always satisfied for any point Q′ of the spiral ‘forward’ of P, so that (2) is also satisfied, and QQ′ is always less than QF.

It will be observed that no νєσις, and nothing beyond ‘plane’ methods, is required in the above proof, and Pappus’s criticism of Archimedes’s proof is therefore justified.

Let us now consider for a moment what Archimedes actually does. in Prop. 8, which he uses to prove our proposition in the ‘backward’ case (R′, R, F′), he shows that, if PO : OV is any ratio whatever less than PO : OT or PM : MO we can find points F′, G corresponding to any ratio PO : OV ′ where OT < OV ′ < OV, i.e. we can find a point F′ corresponding to a ratio still nearer to PO : OT than PO : OV is. This proves that the ratio RF′ : PG, while it is always less than PM : MO, approaches that ratio without limit as R approaches P. But the proof does not enable us to say that RF′ : (chord PR), which is > RF′ : PG, is also alwaysless than PM : MO. At firstsight, therefore, it would seem that the proof must fail. Not so, however; Archimedes is nevertheless able to prove that, if PV and not PT is the tangent at P to the spiral, an absurdity follows. For his proof establishes that, if PV is the tangent and OF′ is drawn as in the proposition, then

or F′O < OR′, ‘which is impossible’. Why this is impossible does not appear in props. 18 and 20, bat it follows from the argument in Prop. 13, which proves that a tangent to the spiral cannot meet the curve again, and in fact that the spiral is everywhere concave towards the origin.

Similar remarks apply to the proof by Archimedes of the impossibility of the other alternative supposition (that the tangent at P meets OT at a point U nearer to O than T is).

Archimedes’s proof is therefore in both parts perfectly valid, in spite of any appearances to the contrary. The only drawback that can be urged seems to be that, if we assume the tangent to cut OT at a point very near to T on either side, Archimedes’s construction brings us perilously near to infinitesimals, and the proof may appear to hang, as it were, on a thread, albeit a thread strong enough to carry it. But it is remarkable that he should have elaborated such a difficult proof by means of props. 7, 8 (including the ‘solid’ νєσις, of Prop. 8), when the figures of props. 6 and 7 (or 9) themselves suggest the direct proof above given, which is independent of any νєσις.

P. Tannery,² in a paper on Pappus’s criticism of the proof as unnecessarily involving ‘solid’ methods, has given another proof of the subtangent-property based on ‘plane’ methods only; but I prefer the method which I have given above because it corresponds more closely to the preliminary propositions actually given by Archimedes.

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¹ Bulletin des sciences mathématiques, 1895, Part i, pp. 265–71.

² Tannery, Mémoires scientifiques, i, 1912, pp. 300–16.