^ip with the Geometry of Boethhts. ,3 ^|^ 537 



with the Arcerian suggested, by the examination of his personal 

 history, the Geometry itself furnishes evidence almost amounting to 

 demonstration, that its author was unacquainted with it. The 

 most important, and, in an historical point of view, the most in- 

 teresting proposition of the mathematical part of the manuscript, so 

 far as its contents are known, is the general formula for any triangle 

 in terms of its sides (p. 301, 11-301, 5)*. Now there is not the 

 slightest hint to be found in any of Gerbert's writings of his ac- 

 quaintance with this formula ; and as we know from his letter to 

 Adelboldf , that his attention had been pointedly directed to the rules 

 then ordinarily used for determining the areas of triangles, it is 

 highly improbable that he should have omitted all mention of it, if 

 it had ever come under his notice. The only rule applicable to 

 all triangles given by hira is, substantially, that the area is equal 

 to half the sum of every side multiplied by the perpendicular let 

 fall on it from the opposite vertex :f. On the other hand, the extract 

 from Hyginus (p. 188, 14-190, 12), with which the Geometry ends, 

 has been taken, not from the Arcerian, but from the Gudian or some 

 other MS. of the second class ; for not only does it agree with 

 the latter, where it differs materially from the first and third class 

 MSS., but also faithfully copies its peculiar blunders and cor- 

 ruptions §. 



The next argument is, that Rigaltius has edited from a MS. of 

 Gerbert's Geometry what is in fact a part of the Demonstratio : and 

 Blume refers to Rigaltius's note in p. 240 : — " Gerbertus, sive quis 

 alius Boetii Geometrica sublegit, postquam ad hujusmodi negotia 

 pervenit, de iis sese nihil attingere velle profitetur :" and he then 

 gives the sentence which has been before quoted from tlie Harleian 

 and Arundel MSS. This certainly creates a difficulty, which, in the 

 absence of more accurate information as to the MS. used by Rigal- 

 tius, it is not easy to overcome. It must be observed that this sen- 

 tence does not occur in the Salzburg MS. of Gerbert ; and in the 

 Arundel, which has a fragment of his Geometry, it forms a part of 

 the Boethius, and not of Gerbert. 



The last argument is derived from the Geometry of Gerbert con- 

 taining the identical extract from Hyginus as to the methods of 

 ascertaining the true direction of the meridian by observations of the 

 sun, which we find in some MSS. of Boethius. Though this ar- 

 gument is apparently entitled to greater weight than the rest, yet jt 



* This formula is found also in the MSS. of Boethius, and has heen published 

 from the second Berne, by Venturi, ' Commentari sopra la Storia et le Teorie dell' 

 Ottica,' p. 125. It agrees with the Excerpta Rostochiensia, when this differs from 

 the Arcerian, except in 300, 1 1, where it has id est, a reading peculiar to itself. 



•f Gerbertus ad Adelboldum de Causa Diversitatis Arcearum in Trigono Equi- 

 latero geometrice arithnieticeve exposito, in Fez. 1. c. 83. 



X See the passages in Fez, 31 and 59. In 61, the area of an equilateral triangle 



, . '*^ — (I 



is said to be equal to — -— . 



§ It is much to be wished that we had some information as to the readings of 

 the Boethian MSS. of this passage. Unfortunately my attention was not directed 

 to this point wlien examining the Cambridge MS. ^,,: j,,.,, d- 



Phil, Mag, S. 3. N9, 246. Suppl, Vol. 36. 2 N 



