84 PROCEEDINQS OF THE [Oct., 1875. 



for lines — 1, 4-1, +2, +3, add +3, +7, +9, +11, respectively, and tlie re- 

 sults will be found in general very fair approximations to the corresponding 

 numbers of the Table where they exist, with five exceptions V, Cb, Cr, Mo, and 

 Sn. If we also except line +3, which has but two members not forming a 

 proper Group, then the algebraic sum of the errors on the remaining 32 atomic 

 weights wlQ be +0. 1, the errors nowhere exceeding +2 or — 2. 



The "parallelism" of the atomic weights pointed out by Dumas, so far as it 

 really exists, may readily be recognized in almost any two lines of the Table, 

 whether adjacent or not, and it is evidently included in, and expressed by the 

 mode of approximating to these numbers just given in paragraph immediately 

 preceding. As easily also can be seen the ' ' pairing of the elements " mention- 

 ed by a writer in the Chemical News, which pairing is a • ' parallelism " with a 

 difference not constant but small, extending over the whole series of per- 

 issads, and such artiads as nearly equal them, the latter greatly outnumbering 

 the former. 



Those who, with Mercer in 1858, and a writer in the Chemical News, 1869, 

 prefer the form of a diagram, may draw a pair of rectangular axes, as in the 

 subjoined figure, lay off on the horizontal one, right and left from the vertical 

 one as positive and negative abscissas, the numbers a in the first column of our 

 Table which mark the atomic equivalences, draw vertical ordinates through 

 each point so found, and mark on these ordinates points, at heights, taken 

 from a scale, corresponding to the atomic weights of the elements in the 

 Group proper to each ordinate. The points indicating the members of each 

 Series, will now be found to He very approximately on a right line, these lines 

 cutting the axis of ordinates at nearly the same angle. The ' ' parallehsm " 

 will now be not only arithmetic but geometric, presented to the eye, and will 

 be most striking in the case of series A, B, C, less so in case of D, E, 

 and of F, G, K Such a diagram may be found useful in instruction, and it 

 will be convenient to take a unit for line of abscissas 10 to 15 times as great as 

 that for the ordinates, giving about 10° for angle formed with axis of abscissas 

 by lines of series A, B, C, and 15° for series G. That the angle will not be pre- 

 cisely the same for all the series is evident from the coefficients of « in a pre- 

 ceeding paragraph, 2 for series C, 2i for E, 3 for G, and also from the differ- 

 ences of the numbers 0, 16, 32, 55, &c. , just given above. 



The remarkable continuity of the three series A, B and C, with the equahty 

 already remarked in the numbers corresponding to a=+4: in one series, and to 

 a= — 4 in the next, permits the construction of another geometric diagram, less 

 convenient than the last for reference, but allowing the three series to be exhi- 

 bited in continuity as one, that is by the use of an Archimedean Spiral whose 

 radius vector increases by 16 units in one revolution. Draw eight hnes radia- 

 ting from one point and including 45° between each pair of rays, mark the 

 extremity of one of these, suppose the downward one, with index 0, the three 

 on the right with +1, +2, +3, successively, the three on the left, with — 1, 

 — 2, —3, successively, the upward one with ±4. Mark on ray +1, by a scale, 

 the points 7, 23, 39 corresponding to Li, Na, K, on ray +2, the points, 9, 3, 

 24, 40, on +3, mark 11, on ± 4, mark 12, 28, on —3, mark 14, 31, on —2, 



