(9) on Hydraulics and the armillary Astrolabe, according to an Arabic compilation, now in the Bodleian (Cod. Arab. CMLIV.). The following also are probably Heron's, (10) katoπTρiká, cited by Damianus who was not much later than Ptolemy. This is probably the same work as the κатожтρiкά printed at Venice, 1518, and then ascribed to Ptolemy. (11) Пepì dióπτpas, on a kind of theodolite. This is ascribed to Heron by the MSS. and was certainly written at Alexandria. It has been edited by M. A. H. Vincent'. (12) Scholia on etc., but some more useful as a fireengine (27), a self-trimming lamp (33), a new kind of cupping-glass (56), a water-clock (63), two small organs (76 & 77). In most of these, the action depends on a vacuum into which water will flow. But no. 50 is a toy in which a metal sphere, filled with steam, is made to revolve by the action of the steam as it issues from two bent spouts fixed in the sphere. (Compare also no. 70). Heron does not claim all the discoveries as his own, and it is curious that Vitruvius (1x. 8 & x. 7) and Pliny (VII. 38), describing similar inventions, attribute them to Ctesibius and say nothing of Heron. The preface shews clearly that Heron did not understand the pressure of the air as causing the filling of the vacuum, but ascribed this result to nature's abhorrence. 1 Text and translation in Notices et Extraits des MSS. de la Biblioth. Impér. Vol. XIX. Pt. II. Paris, 1858, p. 157 sqq. The book contains 33 props. of which the last is the first of the βαρούλκος. The others are of the following kind (1) to find the difference of level between two points, (13) to cut a straight tunnel through a hill from one given point to another, (14) and (15) to sink a vertical shaft to meet a horizontal tunnel, (24) to measure a field without entering it. The dioptra was a straight plank, eight or nine feet long, mounted on a stand but capable of turning through a semicircle. It was adjusted by screws, turning cogwheels. There was an eyepiece at each end and a water-level at the side. With the dioptra two poles, bearing discs, were used, exactly as by modern surveyors. Two appended props. (34) and (35) describe a hodometer, an arrangement of cog. wheels attached to a carriage, so that eight revolutions of the wheel turn the first cogwheel once and the motion is then slackened down through a series of cogwheels of which the last moves a pointer on a measured disc. The proposition from the ẞapoûλкos also describes a machine consisting of a series of cogwheels, started by a screw. The case supposed is that a power of five talents is to move a weight of 1000. In Pappus vi. 10 (Hultsch p. 1061) the power is four talents, the weight 160, and the wheels are of a less diameter. Vincent, who is later than Martin, thinks that there was no Heron of Constantinople at all, but that some writer produced a geodesy, founded on the Dioptra, which he called "a Heron", as we might say "an Euclid". He also remarks that Heron (p. 163, n.) is not a Greek name but in Egyptian "porte une signification qui revient à celle d'ingénieur ". an Euclid, mentioned by Proclus. It exists probably in Arabic at Leyden. (13) Meтρiá mentioned by Eutocius, at the end of his commentary on the Measurement of the Circle, as authority on the extraction of square roots. Parts of this work were (α) τὰ πρὸ τῆς ἀριθμητικῆς στοιχειώσεως (lost), (b) τὰ πρὸ τῆς γεωμετρικής στοιχειώσεως, which is also lost, but portions of which have been preserved in the ὅροι, (c) εἰσαγωγαὶ τῶν γεωμετρουμένων, parts of which are preserved in the γεωμετρούμενα, γεωδαισία, or γεωμετρία, περὶ μέτρων or στερεομετρικά, and γεηπονικὸν βιβλίον, (α) εἰσαγωγαὶ τῶν στερεομεтρоvμévν of which fragments are contained in a work of the same title and also in the last two books mentioned under (c). All these fragments are extant in MS. at Paris and most of them contain tabular statements, made at different dates but all later than our era, of weights and measures. These abridgements and compilations seem to have passed through more than one hand and were made at different dates. The γεηπονιKòv seems to be as late as the 10th century and to have been made at Constantinople. All the works here mentioned which are of mathematical importance were collected and edited in 1864 by Dr F. Hultsch, the well-known authority on ancient metrology and mathematics. Hultsch's volume contains the opot, or Definitions of geometrical names, with a table of measures appended, the yewμerpía, which begins with similar definitions and measures, the γεωδαισία, the εἰσαγωγαὶ τῶν στερεομετρουμένων, Stereometricorum collectio altera, the μετρήσεις οι περὶ μέτρων, the Yenπovikóv, which again has similar definitions and measures, and an extract from the Dioptra on the measurement of triangles'. But no two MSS. contain exactly the same collection, and the contents of these works shew fully the grounds of Martin's opinion upon them. The Heronic formula for the area of triangles is given in the Dioptra and the Geodesy: the Geodesy is practically the same as a large part of the Geometry : the two books on Stereometry contain much repetition of one 1 He adds also Didymi Alexandrini Mensurae marmorum et lignorum and Variae collectiones ex Euclide, Herone etc. another, and the Measurements reproduces all the preceding in a very confused manner. On the other hand, in the Geometry the area of a pentagon is said to be the square of the side × 1, and "elsewhere"" to be the same square and there are other similar discrepancies which point, at the very least, to two editions of the original, if not to gross interpolations and unauthentic additions. The probability is, as Martin suggests, that all these fragments formed part of one comprehensive work on all the knowledge necessary for land-surveying, from which subsequent compilers took, correctly and incorrectly, such matter as they required for their immediate purpose. These extracts in passing from hand to hand, were annotated by many generations of surveyors and thus contradictory statements and extracts from such a late writer as Patricius and references to Roman measures' became incorporated in the text. 145. The character of the contents of the Heronic collection may be indicated in a very few lines. The pot contains 128 definitions of all manner of geometrical terms, followed by a short table of measures. The Geometry begins with a few definitions, followed by an account of the empirical origin of the science, then more definitions, then measures, and passes finally to the solution of problems to find the areas or some linear measurements of triangles, circles, parallelograms and polygons, of which the necessary linear measures or areas are given. The Geodesy, a short extract, begins in the same way 66 1 Elsewhere" is ἐν ἄλλῳ βιβλίῳ τοῦ "Hpwvos, not named, Geom. c. 102, p. 134 (Hultsch), A similar alternative is given on the same page for the hexagon. So on p. 115 the value π=22 is attributed to Euclid, on p. 136 to Archimedes, and this value is generally used, but in the Measurements π=3 is alone used. So, again, although Heron is cited by Eutocius as an authority on square-roots, in the extant works the roughest approximations are continually used. (1) "Let there be a circle with circumference 22, diameter 7 oxowia. To find its area (eußadóv). Do as follows. 7 × 22=154 and 154-38. That is the area. (2) An alternative method, (22) is then added. Then (3) "If you wish to find the area from the circumference only, do as follows. (22)2 × 7=3388 and 38=38." Then (4) to find the area from the diameter only. (5) The same according to Euclid. (6) To find the circumference from the diameter etc. All these examples are applied to circles of various given circumferences, diameters, or areas. Heron then treats similarly of semi but deals only with the areas of given triangles. The Stereometry I. has no definitions but plunges at once into problems to find the volume of given spheres, cubes, obelisks, pyramids and similar figures and next the contents of cups, theatres, diningrooms, baths, etc. The Stereometry II. is chiefly concerned with the same matter as the last part of the preceding book, but in c. 31 (p. 180) suddenly the method of finding heights by measuring shadows is inserted. The Measurements and Geëponicus are a miscellaneous collection of problems similar to or identical with those in the preceding books. The reader will see at once that Heron is chiefly engaged in arithmetical calculations which depend on geometrical formulae, which for the most part he does not, and has no occasion to, prove. Sometimes, however, he actually works out a geometrical theorem. Thus, in the BeλоTоika', he happens to suggest a method of increasing threefold the power of a catapult. This requires that a certain cylinder should be trebled and, as cylinders are to one another as the cubes of their diameters, we are face to face with a problem of triplication of the cube. Upon this, Heron inserts a solution of the duplication-problem, which is identical with that attributed above to Apollonius. In the last chapter of the Geodesy (p. 151), he gives a general formula for finding the area of a triangle. The sides being a, b, C, he says the area is But he works out the proof in the Dioptra. It is as follows. circles, and segments greater or less than a semicircle. On p. 133 occurs the problem, "Given in one number the diameter and the circumference and the area of a circle, to find each." This of course leads to a quadratic equation, of which the solution was given above p. 106. 1 Vett. Math. p. 142. The same proof is given by Pappus (111. p. 63) and Eutocius (in Torelli, p. 136). The latter says it occurs in the μηχανικαὶ εἰσαγωγαί as well as βελοποιϊκά. 2 Reprinted by Hultsch (pp. 235237) who thinks it is interpolated in the Dioptra. The formula, together with Heron's example of its application to a triangle whose sides are 13, 14, 15 (and therefore its area 84), was stolen bodily by Brahmegupta. See Colebrooke pp. 295 sqq. and comments by Vincent op. cit. pp. 200-293, Chasles Aperçu, Note XII. pp. 429 sqq. Cantor pp. 550 sqq. 'Let aßy be the given triangle. Inscribe in it the circle de, having its centre n. Join na, n8, ny, nd, ne, ns. (Comp. Eucl Iv. 4). The rectangle By. ne is double of the triangle ßŋy, and aß.nd of anß, and ay.ns of ayn. Therefore the rectangle under γε and the perimeter of aßy is double the area of aßy. Produce yẞ to 0. Make 60=ad. Then Oy is half the perimeter. Therefore the rectangle Oy. en is equal to the area of the triangle αβγ. Drawn at right angles to my, and Bλ to ẞy and join yλ. Then, the angles yŋλ, yßλ being two right angles, the quadrilateral yn is in a circle. Therefore the angles înß, yλß are equal to two right angles and also equal to the angles ynß, and, which also two right angles (since the angles at ʼn were bisected by an, Bn, yn). Therefore the angle and = angle AB, and the triangles and, yλ are similar. Therefore βγ : βλ :: αδ : δη :: θβ : ηε, and permutando (ἐναλλάξ) By : BO :: Bλ : ŋe :: Bk : ke, and componendo (ovvlévti) yo: 0ẞ:: BE: Єk, and y02: y0. OB :: Be. ey : ey.ek (or ne2). Therefore yox ne2=y0.0ẞ x Be.ery. But ye.ne, which is equal to the area of the triangle, is the square-root (πλevpá) of |