KLM. Now angle A EH being obviously half the angle of an equilateral triangle whose base would be E F, it is at least as great as angle LKM; and, therefore also, because E H exceeds E A, while the angle EAG being obtuse, and EHG a right angle, every straight line drawn from E to meet A G or G H, will exceed E A, since it will be the greatest side of a triangle having either EA or E H for another side; so that the quadrilateral AEHG will be more than capable of containing the triangle K LM, and so will the equal quadrilateral CF HG. It is also evident that by continually increasing DE, DF by parts each equal AE, and drawing other lines to join their extremities, the triangle EDF may be increased to exceed any given area, for it would thereby acquire an unlimited number of increments each greater than area K L M. But if one triangle had the sum of its angles less than 180° by the nth part of that quantity, another triangle whose area is n times as great, would (Prop. II.) have no angles at all, which is absurd, and more especially in the present case, where angle A D C is considerable.

Such being a deduction from the premises in Prop. II., shews that the angles of a triangle can never be less than 180°, which is equivalent to proving Euclid's 12th Axiom, the one being so easily deduced from the other. But if one triangle had its angles greater than 180° by an nth part, another whose area is half of n times as great, would (Prop. H.) have its angles equal to 270°; and, consequently, some two of them would amount to at least 180°, which (Euclid I. 17) is impossible, Hence, there is no triangle whose angles exceed 180°; and, therefore, the three angles of every triangle are just equal to 180°.

The equilateral triangle in Fig. 5 might have been omitted, and a right angle used for ADC, but that would scarcely have conduced to greater brevity or simplicity. If anything could be reckoned to have been assumed above, it would be that the sum of the angles of a triangle can never be inappreciable or so small that it could not be multiplied to exceed any given angle. But if this cannot be regarded as already selfevident, yet the assumption absolutely necessary here to complete the proof would be smaller than any that could be assigned; and, therefore, Euclid's assumption of the angles of a triangle amounting to 180° would exceed the amount required here, in a ratio greater than any that could be assigned. Thus, whatever part of the 180° were supposed to be necessary here as the least sum of the angles; for example, though it were only the millionth, billionth, or any other assigned part, a still smaller would obviously suffice. Under the anticipated objection, therefore, the question would still be reduced within limits incomparably narrower than, so far as I am aware, it had ever been before; especially since most of the authors who have attempted it, assume more than even Euclid assumes.

I shall now endeavour to examine the professedly direct demonstra tions depending on infinite quantities, as employed by Bertrand, in his Developpement de la Partie Elementaire des Mathematiques, and by Le

gendre, Memoires de l'Academie, tome xii. p. 367, (an elaborate essay, but seems to have escaped the notice of Col. Thompson). It will be found that these illustrious geometers have assumed the whole that was to be proved. The following is a theorem of Legendre:-If the indefinite straight lines A C, BD, (Fig. 6) be both perpendicular to A B, and if from any 'point N in BD we draw N M perpendicular to A C, then NM will be equal A B and perpendicular to B D.

Draw NI bisecting A B in I and meeting A C in P, and produce it

Fig. 6.

G

B

N

Y

D

till NQN P. Draw also the indefinite line QY, making angle NQY= PND, and produce M N to meet Y Q in G. From these premises, M. Legendre readily proves that the triangles PIA, BNI are equal, and, consequently, that area D BAC= DNPC; and that angle CPN + PND= DNQ NQY= CPQ + PQY= 180° As readily does he prove the triangles P NM, NGQ to be equal, and therefore area Y Q P C = Y GM C.

I

PAM

C

When any two indefinite straight lines-such as ND, P C-make, with the same side of a third straight line, PN, the two interior angles CPN, PND equal to 180°, Legendre calls the figure DNPC & biangle. Now it is evident that in the above construction we have other four biangles. But, unfortunately, through some sad oversight, M. Legendre in effect makes it to contain one or two besides. Thus, without having proved anything whatever regarding the value of angle M ND, he not only calls CMND a biangle, but reasons upon it as such, or as having two right angles, which obviously is the same as just at once assuming MND = 90°; and it is on this ground he concludes that MNA B. Then, by reasoning in a circle from this conclusion, he proves what, as just stated, he had already assumed, namely, that M N D = 90°. The direct demonstration of Legendre is, therefore, a total

failure.

But since area CMND is less than DNQY by the sum of the triangles P NM, GNQ, the angle G N D cannot be acute; because then CM ND, instead of being less than DNQ Y, would exceed it by the infinite area of an angle equal to the excess of angle M N D over G ND, which is absurd. Were reasoning upon infinite quantities liable to no objection, this absurdity obviously would amount to an indirect proof that the angles of the quadrilateral A B N M cannot be less than 360°, or that those of a triangle cannot be less than 180°.

MM. Bertrand and Legendre regard a biangle as an infinite area of the first order, and maintain that it bears no proportion to, and so could never by repeated subtraction exhaust the area of an angle, which they, reckon an infinite of the second order. But I shall now shew that this doctrine, which is the foundation of Bertrand's demonstration, can only be maintained on the assumption that the angles of a triangle amount to

180°, or those of a quadrilateral to 360°, which is just the whole affair to be proved. For let EFH (Fig. 7) be a right angle, and HFKL á

K

F

V

Fig. 7.

R

I

biangle every way equal to CABD (Fig. 6) and which will also be equal to area C M N D, if, as Legendre assumes, the angle MND 90°. In that case, too, the area E KL would be just as great as the original area EF H, and the rest of the process would seem equally satisfactory.

But if angle BN M be acute, and of course MND obtuse, it is easy to shew how an area equal

to CM ND, or one much less, may be taken from E FH, so as to diminish that infinite area, and, by repetition, exhaust it altogether. Thus, whether F K be taken as great as M N, or in any ratio less, if we only make angle E K L equal to the acute angle G N D, the area H FKL may be equal to or much less than CM ND, and yet the area E KL will evidently be less than area EFH in the same ratio that the acute angle EKL is less than EF H. Draw F V, making angle HFV= LKE; then the sum of the angles H F V, F V L will evidently exceed the sum of HF K, FKL by whatever the three angles of the triangle FV K want of 180°; for if in one case the angles are deficient, so (Prop. II.) they must in every other. If, therefore, K O be taken of any magnitude not exceeding F V, and angle K OR be made equal F V L, the difference between angle EOR and EKL will exceed the difference between E KL and EFH by whatever the three angles of F V K want of 180°. Hence, by continuing the like construction, the residuary angles which the upper sides of the successive lines make with E F, will continually decrease with increasing differences, till they vanish altogether, and thereby exhaust the whole area EF H. Thus, Bertrand's famous demonstration fails entirely, unless we first assume that the angles of every triangle amount to at least 180°.

Another demonstration by Bertrand, and which, like the two just noticed, is highly commended by Legendre, amounts to this: Since the entire indefinite area around a point A, only differs from the sum of the infinite areas of the exterior angles D AE, EBF, FCD of any Fig. 8.

E

-Н

-I

triangle A B C, by the finite area of that triangle itself; which is supposed to bear no proportion to the indefinite area of the plane; it is, therefore, concluded that the sum of the angles around the point A is just equal to the sum of these exterior angles. But such reasoning, in this case, is as faulty as in the two preceding, where the doctrine that a finite quantity has no effect on an infinite, was found to fail, unless we first assumed the very thing to be proved, that the angles of a triangle cannot be less than 180°. For if they may be less, produce A C

D

towards G, and make angle EBHEAG. Then the two angles GCB, CBH will exceed 180° by whatever the three angles of the triangle ABC fall short of 180°; and, consequently, after the finite area A B C has been taken from the infinite E A G, instead of the remainder EBCG being still equal to or less than area E A G, it will exceed area EBH = E AG, by more than the infinite area of an angle H BI equal to the defect of the three angles of A B C from 180°.

This absurdity, while it shews the fallacy of the direct reasoning, would amount, as in the former case, to an indirect proof that the angles of a triangle cannot be less than 180°, were it not that the reasoning still involves infinite quantities, which we have just seen are apt to mislead the greatest of geometers.

On the Light thrown on Geology by Submarine Researches ; being the Substance of a Communication made to the Royal Institution of Great Britain, Friday Evening, the 23d February 1844. By EDWARD FORBES, F.L.S., M.W.S., &c. Prof. Bot. King's College, London. Communicated by the

Author.

About the middle of the last century, certain Italian naturalists* sought to explain the arrangement and disposition of organic remains in the strata of their country, by an examination of the distribution of living beings on the bed of the Adriatic Sea. They sought in the bed of the present sea for an explanation of the phenomena presented by the upheaved beds of former seas. The instrument, by means of which they conducted their researches, was the common oyster-dredge. The results they obtained bore importantly on Geology; but since their time, little has been done in the same line of research, the geologist has been fully occupied above water, and the naturalist has pursued his studies with far too little reference to their bearing on geological questions, and on the history of animals and plants in time. The dredge, when used, has been almost entirely restricted to the search after rare animals, by the more adventurous among zoologists.

Convinced that inquiries of the kind referred to, if conducted with equal reference to all the natural history sciences,

* Marsili and Donati, and after them Soldani.

and to their mutual connection, must lead to results still more important than those which have been obtained, I have, for several years, conducted submarine researches by means of the dredge. In the present communication, I shall give a brief account of some of the more remarkable facts and conclusions to which they have led, and as briefly point out their bearings on the science of geology.

I. Living beings are not distributed indifferently on the bed of the sea, but certain species live in certain parts, according to the depth, so that the sea-bed presents a series of zones or regions, each peopled by its peculiar inhabitants. Every person who has walked between high and low water-marks on the British coasts, when the tide was out, must have observed, that the animals and plants which inhabit that space, do not live on all parts of it alike, but that particular kinds reach only to certain distances from its extremities. Thus the species of Auricula are met with only at the very margin of high water mark, along with Littorina cœrulescens, and saxatilis, Veluting otis, Kellia rubra, Balani, &c.; and among the plants, the yellow Chondrus crispus (Carrigeen, or Iceland moss of the shops), and Corallina officinalis. These are succeeded by other forms of animals and plants, such as Littorina littorea, Purpura lapillus, Trochi, Actinea, Porphyra laciniata (Laver, Sloke), and Ulva. Towards the margin of low water, Lottia testudinaria, Solen siliqua, and the Dulse, Rhodomenia palmata, with numerous Zoophytes and Ascidian molluscs, indicate a third belt of life, connected, however, with the two others, by certain species common to all three, such as Patella vulgata, and Mytilus edulis. These sub-divisions of the sea-bed, exposed at ebb-tide, have long attracted attention on the coasts of our own country, and on those of France, where they have been observed by Audouin and Milne Edwards, and of Norway, where that admirable observer Sars has defined them with great accuracy.

Now this subdivision of the tract between tide-marks into zones of animal life, is a representation in miniature of the entire bed of the sea. The result of my observations, first