24 



NA TURE 



\Nov. 9, 1871 



Notes on the Food of Plants. By Cuthbert C. Grundy, 



F.C.S. (London: Simpkin, Marshall, and Co., 1S71.) 

 This is a useful elementary sketch of the form and manner 

 in which food is obtained by plants. Faults in it there 

 are. Thus, notwithstanding the conclusive experiments 

 of Prillieux and Duchartre, proving that plants have no 

 power of absorbing moisture through their leaves, and the 

 author's own reference to this now established fact in 

 the preface, we still find the assertion (p. 14) that " the 

 leaves withdraw from the atmosphere aqueous vapour." 

 The statement (p. 25) that the sap descends in dicoty- 

 ledonous plants IhroKi^h the bark is not strictly correct ; 

 and a Fellow of the Chemical Society ought not to have 

 described (p. 23) carbonic acid as "carbon dioxide com- 

 bined with water." These blemishes apart, this little book 

 may be recommended to those who desire an explanation 

 of the mode in which vegetable organisms are built up 

 from inorganic materials, and who are unable to devote the 

 time to the more elaborate works of Mr. Johnson, " How 

 Crops Grow" and " How Crops Feed." The portion re- 

 lating to the effect on crops of different soils strikes us as 

 the best. 



LETTERS TO THE EDITOR 



\The Editor docs not liold himself n'sponsible for opinions expressed 

 by his correspondents. No notice is taken of anonymous 

 commnnications. ] 



Proof of Napier's Rules 

 As the following graphical construction is easily executed, re- 

 presenting to the eye the figure usually employed for the proof 

 of Napier's rules of the pirts of right-angled triangles in 

 spherical geometry, it will perhaps remr]ve diffi--uUies from their 

 proof for beginners, like those which Mr. W. D. Cooley's work 

 on "Elementary Geometry " must, from his description of some 

 interesting parlsof its contents in Nature of the 19th of October, 

 have proposed to itself to meet, and to render at least as easily 

 accessible as possible to the inquiring student in mathematics. 



BF is a rectangular card, measuring two inches by three inches 

 in the sides, and divided by the lines DB, DC, D.\, DB', and 

 B'C in the directions shown in the figure, and in such a manner 

 that the three corners of the rectangle are completely cut away 

 by the last two, and by the first of these lines ; while DC and DA 

 are only cut or scored lightly in the card, so as to allow the re- 

 maining three triangles, DBC, DCA, DAB', to be folded towards 

 each other, until, DB and DB' coinciding, they form a solid angle 

 of three faces at the point D. The properly possessed by this 

 solid angle, that the inclination of the two faces, DCB, DCA, 

 to each other is a right angle (the angle shown at C in the base, 

 AB'C of the solid angle), .and that the base AB'C of the result- 

 ing tetrahedron cuts the two faces ADC, ADB', perpendicularly 

 (or at right angles to their common intersection DA) in the line 

 AC, AB', so tliat the plane angle A of the plane right-angle 

 triangle B'AC is also the inclination between those faces, or t.c 



angle of the right-angled spherical triangle formed by the inter- 

 section of a sphere, about the centre D, with the three planes 

 meeting each other at that point, affords a ready proof of all 

 Napier's rules, excepting that connecting the two angles of a 

 right-angled spherical triangle, from the simple definitions of the 

 trigonometrical " ratios" of plane angles.* 



Calling the angles of the faces which meet together at the point 

 D, as shown in the figure <r, /', c, opposite to the spherical angle-; 

 A,B,C, formed by the inclination of the other two faces to each 

 other, these angles, and those of inclination of the faces are, re- 

 spectively, ihe sidrs and angles of a right-angled spherical 

 triangle, whose right angle is C, its hypothenuse is c, and the 

 angle A, between /' and c is equal to the plane angle A, of the 

 right-angled triangle AB'C. 



Taking, firstly, as the radius, DA, equal to unity, AC (or 

 AC), and AB' are the tangents of h and <- ; and the right- 

 angled triangle AC'B' gives the rule. 



t^" ''' A A . ; 



— = cos A ; or cos A = tan l< 



tan < 



cot . 



(«) 



cos A -H sin : 



cos a, by (3) ; 



Taking, in the next place. DB, (or BB'), as the radius equal 

 to unity ; BC (or B'C), and B'A are the sines ; and DC, D.\ are 

 the cosines of the angles a and c. In the first case the right- 

 angled triangle AB'C affords the ratio 



^'" " = sin A ; or sina = sin c . sin A ; (2) 



sin c 



And in the second case we obtain from the right-angled 

 triangle ADC the rule 



cos I' = cos a . cos b (3) 



The rules for the angle B, corresponding to (i) and (2) for the 

 angle A, are simply obtained from them by transposing in them 

 the sides and angles i;A for i^B ; thus — 



cos B = tan a . cot c (4) 



sin b = s>\d c . sin B (5) 



Finally, dividing (l) by (5), a rule for connecting together the 

 two angles of the right-angled spherical triangle is found as 

 follows : — 



tan i , sin 1^ _ cos c 

 tan c ' sin c cos b 

 or cos A = cos a sin B (6) 



If, as in Napier's rules, the two sides and the differences from 

 90° of the two angles and of the hypothenusa arranged in their 

 natural order round the triangle are regarded as constituting its 

 five parts, it will be seen that all the above consequences may be 

 included in the two rules known as Napier's rules, that the sine 

 of the middle (that is, of any chosen) part is equal to the product 

 of the tangents of the two adjacents, as well as to the product of 

 the cosines of the two opposite parts. 



As a rule to assist the memory, the laconic brevity and com- 

 pleteness of Napier's formula possess a most uniquely felicitous, 

 and, happily for mathematicians, a not unfrequently enduring 

 charm. But should the student desire to divest himself of their 

 artificiality, and to retrace for himself the steps of the demonstra- 

 tion upon which any one example of these rules is based, he must 

 first draw a solid tetrahedron ABCD, in which the facial angles 

 at A, C, are as represented in the figure, but as they cannot all 

 be correctly shown on account of the embarrassing effects of the 

 perspective in the drawing, right angles. By having recourse to a 

 model, on the other hand, which may very readily be cut from a 

 card like that illustrated in the above description, and folded so 

 as to form the solid figure required for their demonstration, all 

 the cases of Napier's rules may be e.vhibittd, and proved, almost 

 as speedily, and satisfactorily to a learner's apprehension in solid 

 geometry, as the definitions of the simple trigonometrical ratios of 

 plane angles, and the least complicated relations connecting 

 together the parts of plane triangles may be made intelligible to 

 him ; and that by a plain series of immediate deductions from the 

 figure, which his familiarity with the processes of plane trigo- 

 nometry will already have taught him very easily to supply. 



Newcasde-on-Tyne, Oct. 30 A. S. Herschel 



Remarkable Paraselene seen at Highfield House on 

 October 25th, 1871 

 The phenomenon first became visible at yh 12"' p.m., and 

 finally vanished at 7'" 33 P.M. The upper portion of a halo of 



' Another simil.-ir property, with a somewhat less important application of 

 the same tetrahedron, is described in the Quarterly Jourftal 0/ Mil the mat id 

 for October i86z, p. 306. 



