The "new invention in Denmark" to which Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthaphaeresis (often written in Greek letters [Greek: prosthaphairesis]), which had its origin in the solution of spherical triangles.[7] The method consists in the use of the formula sin a sin b = ½{cos (a - b) - cos (a + b)}, by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only. It seems to be due to Wittich of Breslau, who was assistant for a short time to Tycho Brahe; and it was used by them in their calculations in 1582. Wittich in 1584 made known at Cassel the calculation of one case by this prosthaphaeresis; and Justus Byrgius proved it in such a manner that from his proof the extension to the solution of all triangles could be deduced.[8] Clavius generalized the method in his treatise _De astrolabio_ (1593), lib. i. lemma liii. The lemma is enunciated as follows:-- "Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi solent, per solam prosthaphaeresim, id est, per solam additionem, subtractionem, sine laboriosa numerorum multiplicatione divisioneque expedire." Clavius then refers to a work of Raymarus Ursus Dithmarsus as containing an account of a particular case. The work is probably the _Fundamentum astronomicum_ (1588). Longomontanus, in his _Astronomia Danica_ (1622), gives an account of the method, stating that it is not to be found in the writings of the Arabs or Regiomontanus. As Longomontanus is mentioned in Anthony Wood's anecdote, and as Wittich as well as Longomontanus were assistants of Tycho, we may infer that Wittich's prosthaphaeresis is the method referred to by Wood. It is evident that Wittich's prosthaphaeresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations. It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle from the logarithmic method. In fact, if we put xy = [phi](X + Y), X being a function of x only and Y a function of y only, we can show that we must have X = Ae^(qx), y = Be^(qy); and if we put xy = [phi](X + Y) - [phi](X - Y), the solutions are [phi](X + Y) = ¼(x + y)², and x = sin X, y = sin Y, [phi](X + Y) = -½cos(X + Y). The former solution gives a method known as that of quarter-squares; the latter gives the method of prosthaphaeresis.