+--------------------------++-------------------------------------+ | AB 28} × { 84} × { 672 || DG 3} × { 72} × {576 | | BJ 45} 3 {135} 8 {1080 || GE 4} 24 { 96} 8 {768 | | JA 53} {159} {1272 || ED 5} {120} {960 | +--------------------------++-------------------------------------+ | DC 3} × {135} × {1080 || FW 48} × { 48} × {384 | | CA 4} 45 {180} 8 {1440 || WV 55} 1 { 55} 8 {440 | | AD 5} {225} {1800 || VF 73} { 73} {584 | +--------------------------++-------------------------------------+ | EB 3} × { 63} × {504 || FB 20} × { 80} × {640 | | BA 4} 21 { 84} 8 {672 || BA 21} 4 { 84} 8 {672 | | AE 5} {105} {840 || AF 29} {116} {928 | +--------------------------++=====================================+ | FH 3} × { 96} × { 768 || | | HN 4} 32 {128} 8 {1024 || Note.--In the above table the first | | XF 5} {160} {1280 || column _is the Ratio_, the second | +--------------------------++ _the connected Natural Numbers_, and| | AY 3} × { 36} × { 288 || the third column represents _the_ | | YZ 4} 12 { 48} 8 { 384 || _length each line in R.B. cubits_. | | ZA 5} { 60} { 480 || | +--------------------------++-------------------------------------+ Fig. 60. Reference to _Fig. 60_ and the preceding table, will show that the main triangular dimensions of this plan (imperfect as it is from the lack of eleven pyramids) are represented by four main triangles, viz:-- Ratio. C A D C .. .. 3, 4, 5 F B A F .. .. 20, 21, 29 A B J A .. .. 28, 45, 53 F W V F .. .. 48, 55, 73 Figures 30 to 36 illustrate the two former, and _Figures_ 61 and 62 illustrate the two latter. I will call triangles of this class "primary triangles," as the most suitable term, although it is applied to the main triangles of geodetic surveys. We have only to select a number of such triangles and a system of trigonometry ensues, in which base, perpendicular, and hypotenuse of every triangle is a whole measure without fractions, and in which the nomenclature for every angle is clear and simple. An angle of 43° 36′ 10·15″ will be called a 20, 21 angle, and an angle of 36° 52′ 11·65″ will be called a 3, 4 angle, and so on. In the existing system whole angles, such as 40, 45, or 50 degrees, are surrounded by lines, most of which can only be described in numbers by interminable fractions. In the ancient system, lines are only dealt with, and every angle in the table is surrounded by lines measuring whole units, and described by the use of a couple of simple numbers.