Connecting this with our present system of trigonometry would effect a saving in calculation, and general use of certain peculiar angles by means of which all the simplicity and beauty of the work of the ancients would be combined with the excellences of our modern instrumental appliances. Surveyors should appreciate the advantages to be derived from laying out traverses on the hypotenuses of "primary" triangles, by the saving of calculation and facility of plotting to be obtained from the practice. The key to these old tables is the fact, that in "primary" triangles the right-angled triangle formed by the sine and versed sine, also by the co-sine and co-versed-sine, is one in which base and perpendicular are measured by numbers without fractions. These I will call "satellite" triangles. Thus, to the "primary" triangle 20, 21, 29, the ratios of the co-sinal and sinal satellites are respectively 7 to 3, and 2 to 5. (_See Figure 35._) To the 48, 55, 73 triangle the satellites are 11, 5 and 8, 3 (_Fig. 62_); to the 3, 4, 5 triangle they are 2, 1 and 3, 1 (_Fig. 30_); and to the 28, 45, 53 triangle, they are 9, 5 and 7, 2 (_Fig. 61_). The primary triangle, 7, 24, 25, possesses as satellites the "primary" triangle, 3, 4, 5, and the ordinary triangle, 4, 1; and the primary triangle 41, 840, 841, is attended by the 20, 21, 29 triangle, as a satellite with the ordinary triangle 41, 1, and so on. Fig. 61. The 28-45-53 Triangle. Fig. 62. The 48-55-73 Triangle. Since any ratio, however, whose terms, one or both, are represented by fractions, can be transformed into whole numbers, it evidently follows that every conceivable relative measure of two lines which we may decide to call co-sine and co-versed-sine, becomes a satellite to a corresponding "primary" triangle. Now, since the angle of the satellite on the circumference must be _half_ the angle of the adjacent primary triangle at the centre, it follows that in constructing a list of satellites and their angles, the angles of the corresponding primary triangles can be found. For instance-- Satellite 8, 3, contains 20° 33′ 21·76″ Satellite 2, 7, contains 15° 56′ 43·425″ Each of these angles doubled, gives the angle of a "primary" triangle as follows, viz.:-- The 48, 55, 73 triangle = 41° 6′ 43·52″ The 28, 45, 53 triangle = 31° 53′ 26·85″ The angles of the satellites together must always be 45°, because the angle at the circumference of a quadrant must always be 135°. From the Gïzeh plan, as far as I have developed it, the following order of satellites begins to appear, which may be a guide to the complete Gïzeh plan ratio, and to those "primary" triangles in use by the pyramid surveyors in their ordinary work.