Page:The Rhind Mathematical Papyrus, Volume I.pdf/183

1903 and, The Oxyrhynchus Papyri, Part III, edited with Translation: and Notes, (Egypt Exploration Fund, Graeco-Roman Branch), London, 1903.

Number 470 (pages 141-146) in this collection is described as a "mathematical treatise" 16.7 × 19 cm., of the third century A. D.; "a leaf, of which the top is lost, from a papyrus book . . . containing apparently descriptions of astronomical instruments." Lines 31 to the end are concerned with the construction of a water-clock in the shape of a frustum of a right circular cone of which the diameter of the upper section is 24 ﬁngers, of the lower 12, while the distance between the sections is 18 ﬁngers. (The cotangent of the angle which a generator of the frustum makes with its projection on the plane of the lower section (seked?) is, therefore, ⅓, corresponding to an angle of 71° 34′). If the distance, h, between the sections be divided into 18 equal parts each part may be regarded as the altitude (1 finger) of a new frustum. In the papyrus the volumes of the upper six of these frusta are found by calculations equivalent to using the formula (D and d being the diameters of the circular sections): h(πr/3) [(D +d) /2]-(π/4) (D +d) /2 = (⅓) hπ(π/4) (¼) (D$2$ + 2 Dd +d$2$) if π were taken equal to 3 this formula and that for the volume of a right circular cylinder of height h and the diameter of whose base is ½ (D+d) would reduce to the same expression. The correct formula for the frustum is (½) hπ(¼) (D$2$ + Dd + d$2$). The introduction of "(i. e. by π)," p. 145, line 25, is a very misleading comment on the part of the editors.

It is interesting to note that Heron of Alexandria, who probably flourished about the time that this papyrus was written, carried through a numerical problem which seemed to indicate familiarity with the second of these formula, and with the formula h (π/4) (¼) (D + d)$2$; compare Heranis Alexandrini Opera quae supersunt omnia, Leipzig, vol. 5, ed. by Heiberg, 1914, pp. 12-17, and vol. 3, ed. by Schöne, 1903, pp. 116-119; in both cases π is taken as 22/7, and the numerical work of the ﬁrst corresponded rather to the formula for the volume in the form: ¼hπ [(½)$2$ (D + d)$2$ + (⅓) (½)$2$(D—d)$2$].

Papyrus 470, which is now in the Library of Trinity College, Dublin, is reproduced in facsimile in its natural size on plates 7-8 of L. Borchardt, Die Altdgyptische Zeitmessungen (Die Geschichte der Zeitmessungen und der Uhren, herausgegehen von Bassermann-Jordan, vol. 1, part B), Berlin and Leipzig, 1920. The transcription and translation are practically identical with those given by Grenfell and Hunt. The oldest known specimen of a water clock of the above form, from which time was determined by the lowering of the water level, dates from about 1400 B. C. It was discovered at Karnak in 1904 and is now in the Museum at Cairo: it is discussed and pictured by Borchardt, pp. 68—78 and plates 1-3.

D. Limongelli considered, "Note sur une clepsydra antique," Bulletin de L'Institut Egyptien, series 5, vol. 9, pp. 51-52, what shape a clepsydra must have in order that the level of the water be proportional to the time, and found