JORDANUS DE NEMORE (fl. ca. 1220), mechanics, mathematics.

FIGURE 5

Guido Ubaldo's attitude was costly, for it led him to reject Jordanus' correct inclined-plane theorem in favor of an erroneous explanation by Pappus.

Since few editions of his mathematical treatises have been published, and critical studies and evaluations are largely lacking, Jordanus' place in the history of medieval and early modern mathematics has yet to be determined. Treatises on geometry, algebra, proportions, and theoretical and practical arithmetic have been attributed to him.

The Liber Philotegni de triangulis, a geometrical work extant in two versions, represents medieval Latin geometry at its highest level. In the four books of the treatise we find propositions concerned with the ratios of sides and angles; with the division of straight lines, triangles, and quadrangles under a variety of given conditions; and with ratios of arcs and plane segments in the same circle and in different circles. The fourth book contains the most significant and sophisticated propositions. In IV.20 Jordanus presents three solutions for the problem of trisecting an angle, and IV.22 offers two solutions for finding two mean proportionals between two given lines. A proof of Hero's theorem on the area of a triangle--A = ?s(s - a)(s - b)(s - c), where s is the semiperimeter and a, b, and c are the sides of the triangle--may also have been associated with the De triangulis. Jordanus drew his solutions largely from Latin translations of Arabic works, which were themselves based on Greek mathematical texts. He did not always approve of these proofs and occasionally displayed a critical spirit, as when he deemed two proofs of the trisection of an angle based on mechanical means inadequate and uncertain (although no source is mentioned, they were derived from the Verba filiorum of the Banū Mūsā) and offered what is apparently his own demonstration,18 in which a proposition from Ibn al-Haytham's Optics is utilized. In IV.16 a non-Archimedean proof of the quadrature of the circle may have been original. It involves finding a third continuous proportional. Here is the proof.19

To Form a Square Equal to a Given Circle.

For example, let the circle be A [see Figure 5].

Disposition: Let another circle B with its diameter be added; let a square be described about each of those circles. And the circumscribed square [in each case] will be as a square of the diameter of the circle. Hence, by [Proposition] XII.2 [of the Elements], circle A/circle B = square DE/square FG. Therefore, by permutation, DE/A = FG/B. Let there be formed a third surface C, which is a [third] proportional [term] following DE and A. Now C will either be a circle or a surface of another kind, like a rectilinear surface. In the first place, let it be a circle which is circumscribed by square HK. And so, DE/A = A/C but also, by [Proposition] XII.2 [of the Elements], DE/A = HK/C. Therefore, HK as well as A is a mean proportional between DE and C. Therefore, circle A and square HK are equal, which we proposed.

Next, let C be some [rectilinear] figure other than a circle. Then let it be converted into a square by the last [proposition] of [Book] II [of the Elements], with its angles designated as R, S, Y, and X. And so, since DE is the larger extreme [among the three proportional terms], DE is greater than square RY, and a side [of DE] is greater than a side [of RY]. Therefore, let MT, equal to RX, be cut from MD. Then a parallelogram MN--contained by ME and MT--is described. Therefore, MN is the mean proportional between DE and RY, which are the squares of its sides, since a rectangle is the mean proportional between the squares of its sides. But circle A was the mean proportional between them [i.e., between square DE and C (or square RY)]. Therefore, circle A and parallelogram MN are equal. Therefore, let MN be converted to a square by the last [proposition] of [Book] II [of the Elements], and this square will be equal to the given circle A, which we proposed.

The De numeris datis, Jordanus' algebraic treatise in four books, which was praised by Regiomontanus, was more formal and Euclidean than the algebraic treatises derived from Arabic sources. Indeed it has recently been claimed20 that in the De numeris datis Jordanus anticipated Viète in the application of analysis to algebraic problems. This may be seen in Jordanus' procedure, where he regularly formulated problems in terms of what is known and what is to be found (this is tantamount to the construction of an equation), and subsequently transforms the initial equation into a final form from which a specific computation is made with determinate numbers that meet the general conditions of the problem.