. Geometrical and graphical essays, containing a general description of the mathematical instruments used in geometry, civil and military surveying, levelling, and perspective; with many new practical problems. er tables of the northing, southing, east,ing and westing, fitted purposely for the surveyorsuse, such as are contained in the Af>pendix to this7ir^^/, ought to be in the hands of every surveyor. Bythese tables the area of any survey is determined with )32 AN UNIVERSAL THEOREM. great accuracy and expedition, by an easy computa-tion from the following UNIVERSAL THEOREM. If the sum of
. Geometrical and graphical essays, containing a general description of the mathematical instruments used in geometry, civil and military surveying, levelling, and perspective; with many new practical problems. er tables of the northing, southing, east,ing and westing, fitted purposely for the surveyorsuse, such as are contained in the Af>pendix to this7ir^^/, ought to be in the hands of every surveyor. Bythese tables the area of any survey is determined with )32 AN UNIVERSAL THEOREM. great accuracy and expedition, by an easy computa-tion from the following UNIVERSAL THEOREM. If the sum of the distances, on an east and westIhie, of the two ends of each line of the survey, fromany assumed meridian laying entirely out of the survey,le multiplied hy the respective northing 6;south-1N G 77uide on each respective line ; /^^ d i f f e r e x c ebetween the sum of the north vrotivcis, and thesum of the sovTH products, will he double the areaof the survey.* d * Demonstration. Let ABC ^D E F be any survey;, and let ^the east and west lines A a, B b,C c, D d, E e, and F f, bedrawn from the ends of each of ^the lines of the survey to anyassumed meridian, as N. S. lay- ig entirely out of the survey. /. CO ^.J^ Then. [ say, the difTcrcnce between the £um of the products A a-4-B bx a b, B b+C ex b c, C c+D dx c d 5 and the sum of the products Dd+EeXde, Ee-f-Ffxef, Ff+Aaxfa3 will bedouble the area of A B C D E F A. For,by article 4,page327,the former productswill be, respectively,double the areas of the spaces A E b a, B C c b, and C D dc, and theirsum will of course be double the area ot the space A a d D C B like , the latter products will be, respectively, double theareas of the spaces D E e d, E F f e, and F A a f j and their sumwill of course be double the area of the space A ad it is evident, that the difierence between the areas of the spacesA a d U E F A, and A a d D C B A, must be the area of thespace A B C D E F A ; consequently, the ditterence be
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