RECREATIVE NATriiAh HISTORY. 



.'7! 



PROPOSITION XXVIII. If, In tho figure of Eno. I. 

 produced out tho circles again in D, K (Fig. 27), ami the 

 out again in r, tho figure c r D shall bo a rhombnii, having 

 each of the angles at c and F, half the angles at 



.loin A F, r, i. . . i ; then because the angles OA , CAD are 

 .-. ;;iat to two right angles, and similarly o B A and B E (Euc. 

 I. i.".i, therefore angles o A B and CAD are equal to CDA and 

 i' B E. But c A B c B A (Euo. 1. 1), therefore remainder c A D = 

 remainder CUE; and since A D, A c, 

 A D, B c, B E are nil equal, being radii 

 of equal circles, therefore in tl.o two 

 triangles CAD, c B E, because sides 

 c A, A D = sides c B, B E, each to 

 each, and included angle c A D 

 included angle c B K, therefore base 

 C D = base c E (Euc. L 

 an exactly similar process of reason- 

 ing. D F = V E, and because c B = 

 B F, and B D is common, and included angles c B D, r B D ore 

 equal, each of them being the angle of an equilateral triangle, and 

 therefore equal to one-third of two right angles (Euo. 1. ''->, 

 therefore base c D equals base D F, similarly c E = E F ; there- 

 fore CD, D F, F E, EC aro equal, and c D F E is a rhombus. 

 Apain, because A c = A D, tho angles A c D, A DC are equal (Enc. 

 I. 5) ; but angle c A B is equal to the two interior and opposite 

 angles A c D and A DC (Km 1 . I. ."-), therefore the angle c A B is 

 double cither of tho angles Ac D or ADC. Similarly tho angle 

 o it A i.s double either of the angles n c i: or B E c ; therefore the 

 angle A c B, which is equal to cither of the angles c A B or A B c, 

 is double either A c D or A D c, or B c E or B E c ; hence A c B is 

 equal to the sum of A c D and B c E, and is also equal to tho 

 Bum of ADC and EEC; therefore tho whole angle DCE is 

 doable ACB, and is therefore four times the angle c D E or 

 o K D ; similarly, the angle D F E is double A F B, and is therefore 

 four times the angle F D E or F E D. But F D E or F E D is equal 

 to c D E or c E D, hence either of tho angles D c E or D F K is 

 double either of the angles c D F or c E v. Q. E. D. 



Our next article will extend as far us Eac. I. 40, and we shall 

 give solutions of the following propositions : 



PROPOSITION XXIX. To trisect a given right angle, that is, 

 to divide it into throe equal parts. 



PROPOSITION XXX. If two right-angled triangles have one 

 side and tho base in one equal to one side and the base in the 

 other, each to each, they shall be equal in every respect 



PROPOSITION XXXI. The straight lines which bisect the 

 angles of a triangle meet in a point. 



PROPOSITION XXXII. Tho straight lines drawn perpen- 

 dicular to the sides of a triangle through their middle points 

 meet in a point. 



PROPOSITION XXXIII. The straight lines which bisect one 

 interior and two exterior angles of a triangle meet in a point. 



PROPOSITION XXXIV. If two triangles have one side, and 

 one angle in the one equal to one side and one angle in tho 

 other, and likewise their areas equal, then shall also their other 

 sides and angles be equal each to each. 



PROPOSITION XXXV. If the bases of two equal triangles be 

 in the same straight line, and the line joining their vertices be 

 parallel to this line, their bases will be equal. 



PROPOSITION XXXVI. In the figure of Euo. I. 5, if A c bo 

 bisected in H, and c a be equal to c A, then B a shall be equal 

 to twico B H. 



RECREATIVE NATURAL HISTORY. 



SOME LAND, SEA AND FRESHWATER SHELLS, WORMS, 

 AND TUBE-DWELLERS (confining. 



PEW of our readers who have investigated the habits of the 

 deeply-interesting and curious creatures found amongst the rooks 

 and rock pools of our own coast, will have failed to notice tho 

 deep tubular excavationu made in the rock by those accomplished 

 and industrious borers, the Pholas family. In different localities 

 we find two of these stone perforators (Pholas dactyius and Stm- 

 cava rugosa). Tho former of these we find prosecuting hia 

 labours both amongst the chalk rocks and red sandstone of the 

 southern coast of England, whilst the latter, not content with 

 attacking substances of an ordinary degree of hardness, proceeds 

 to operate on the compact, hard limestone rock, cutting hi* way 

 deeply into it, just as a skilful carpenter bores an angur-hole 



in a door-post It i by shelb of tUs Had thaithe bnjpblodn 

 of stone used in building the Plymouth Breakwater and some of 

 the new military works are slowly but surely beinr reduced to a 

 species of stone honeycomb. Not only stone bat solid and 

 dense grained timber is readily bored into by the pVdsvilft We 

 stated in our last paper on this subject that :iooh difference of 

 l and scientific argument had arisen on the subject of the 

 boring powers of these curious creatures. Some philosophers 

 hare stoutly maintained that the animal secreted a fluid of " add " 

 reaction, which possessed the power of so acting on the eosv 

 stituents of wood, stona, amber, wax, and gum resins that they 

 became sufficiently soft snd diiintpgrsted * to admit of the 

 shell, together with the molluttk inhabiting it, pawing freely into 

 and through the imbalance acted on. Others have maintained 

 that tho minute roxp-liko tocth, or asperities, with t : *h shells 

 of this kind are armed, being constantly brought to bear on the 

 exposed surface of the stone at the bottom of the perforation, 

 were alone the agents in force to deepen the tube, atmtlmr set 

 of investigators have stated that the borders of the soft coating 

 or month of tho mollusk, aided by iU short, stoat foot, were 

 the means employed. It has been also urged that the constant 

 and decomposing action o ' minute currents of sea-water passing 

 through tho siphon-like tissues of the animal brought about the 

 fretting action requisite to form a hole. 



Wo hare broken tho pholas shells from stones of a sharp, 

 sand grit, which would be found to grind the hardest steel 

 rapidly away. These shells we hare examined under a powerful 

 ies on them have been by us most carefully 

 scrut inised, but without our being enabled to detect the liflht<st 

 evidence of wear and tear by friction. Every minnt point 

 remained as sharp as a now needle, and bore no traces of having 

 cut through a mass of stone thick enough to hare destroyed the 

 points of a dozen engraving tools. Then when two of these 

 miners so drive their galleries that they intersect 

 each other, tho more powerful workman of the two, ignoring the 

 of his weaker fellow-labourer, works on, bores forwards, 

 and not only tunnels the rock, but tho shells and soft tissues of 

 his neighbour, literally boring him through and through. We 

 have never been able to detect by the ordinary tests any "acid" 

 in the water thrown off from the siphon of the pholas. 



The rock-boring snails (Helix icucicava) before described by 

 us, although forming deep tubular chambers in hard, dense rook, 

 have no currents of sea-water to aid them, neither have they 

 the same rasp-like and rounded character of shell. The nag- 

 like mouth or portal of a snail-shell could but grind and wear 

 down (supposing the file process to be that in force) in an 

 uneven circle corresponding with the shell border. 



To illustrate our point a little more clearly, let us place a com- 

 mon wine-glass or & metal thimble month downwards on a piece 

 of soft Bath-brick, and then proceed to work it round until it 

 penetrates the substance on which it is placed. On *rm*Jmm- 

 tion of our work, we shall find a groove corresponding to the 

 edge of tho circle of friction, and an even, table-like middle no 

 ! deeper than the plane surface of the brick, which, as the groove 

 deepened, would enter the month of the snail-shell, press up the 

 ant, and finally stop his boring operations altogether. 

 Wo find, however, on examining a real snail-tube, that it if. 

 although high and dry on land, formed much like that made by 

 the pholas, and the bottom of the excavation, instead of being 

 even, is cup-like in form, the centre, where no ring-shaped shell 

 could touch, being the deepest point ; and, curiously enough, 

 the tracks made by the snails in going to and from their winter 

 retreats, year after year, are of a distinctly grooved form. Two 

 snails aro not uncommonly found in the same tube, bat, unlike 

 :li pholado.s, they never in any way interfere with each other. 



It has been stated that a distinct acid reaction has, by the aid of 

 litmus paper, been detected in the fluids given off by the boring 

 snail. To this statement we attach but little importance. Visit 

 a nest of black wood-ants, place a piece of slate in a split 

 stick, hold it over the ant-hill, and then irritate the community ; 

 acid enough to act strongly on litmus paper will be at once de- 

 posited on the stone. And yet ants do not bore holes in rocks. 



Tho composition of the secretions of living organisms is, 

 in many instances, entirely beyond the powers of the most 

 accomplished chemist either to imitate or correctly lay down. 

 The silkworm spins for itself a cocoon, or capsule, in which to 

 rest until the period fur change into the moth stage arrives. 

 Examine one of these ooooons, and reflect as to how a tiny, 



