Multiply likewise the expressions of art. 2. and 5., and 7. = sin AC cos ΔΑ If, in all the preceding formula, the increments annexed to the varying quantities be omitted, there will arise much simpler expressions for the differentials. Case II.—When one side a, and its opposite angle ▲, are con stant. a sinB b sinA, and taking the differences by art. 1. of Note 10. Ab sinA = 2a sinдABcos(B+AB), whence sin AB sin A acos(B+4B)' ΔΙ and consequently, by art. 5. of Note 12. 8. sin B ΑΔΙ b cos(B+4B)* In like manner, it will be found that cos(C+AC) The differentials are discovered, by rejecting the modifications of the variable quantities. Case III. When one side a, and its adjacent angle B, are con stant. In the incremental triangle contained by the sides b, b + Ab, and Ac, it is evident, (art. 5. Note 12.), that 11. Ac sinaC Δε = b = 6+4 sinAA sin(A+^A) Again, in the same incremental triangle, (art. 6. Note 12.) 12. tans AC tanž▲▲TM tan(A+1⁄2▲A)° Or, transforming the preceding expression, sin AC 12. (cos(A+{AA cos{^^)) =—sinA§▲A (COS (+14) 146 sin▲A = b (cos (A+AA)). sin(A+▲A) sin(A+AA) Again, in the same incremental triangle, by art. 20. Note cos(A+}AA)=46 (—cos}AC)=cos AA; whence 1 The differentials are found as before, by the omission of the minute excrescences. To compute the values of the finite differences, when these differences themselves are involved in their compound expression, the easiest method is to proceed by repeated approxima tions. Thus, from art. 3. Ac—— (2ccot (A+AA) tan AB cot(A+AA) tan AB (2c+Ac); as cot (A+44A) 2c; and then, ac tan AB cot (A+AA) 2c). But it will sel dom be requisite to advance beyond two steps; though the process, if continued, would evidently form an infinite converging series. When only one part of a triangle remains constant, the expressions for the finite differences will often become extremely complicated. It may be sufficient in general to discover the relations of the differentials merely. To do this, let each indeterminate part be supposed to vary separately, and find, by the preceding formula, the effect produced; these distinct elements of variation being collected together, will exhibit the entire differential. The materials of this intricate Note appear in Cagnoli, but the subject was first started by our countryman Mr Cotes, a mathematician of profound and original genius, in a brief tract, entitled Estimatio errorum in mixtâ Mathesi. It is unfortunate that I have not room for explaining the application of those formula to the selection and proper combination of triangles in nice surveys. 20. HAVING in some of the preceding notes briefly pointed out the several corrections employed in the more delicate geodesiacal operations, I shall subjoin a few general remarks on the application of trigonometry to practice. The art of surveying consists in determining the boundaries of an extended surface. When performed in the completest manner, it ascertains the positions of all the prominent objects within the scope of observation, measures their mutual distances and relative heights, and consequently defines the various contours which mark the surface. But the land-surveyor seldom aims at such minute and scrupulous accuracy; his main object is to trace expedi tiously the chief boundaries, and to compute the superficial contents of each field. In hilly grounds, however, it is not the absolute surface that is measured, but the diminished quantity which would result, had the whole been reduced to a horizontal plane. This distinction is founded on the obvious principle, that, since plants shoot up vertically, the vegetable produce of a swelling eminence can never exceed what would have grown from its levelled base. All the sloping or hypotenusal distances are, therefore, reduced invariably to their horizontal lengths, before the calculation is begun. Land is surveyed either by means of the chain simply, or by combining it with a theodolite or some other angular instrument. The several fields, are divided into large triangles, of which the sides are measured by the chain; and if the exterior boundary happens to be irregular, the perpendicular distance or offset is taken at each bending. The surface of the component triangles is then computed from Prop. 29. Book VI. of the Elements of Geometry, and that of the accrescent space by Note 4. to Prop. 9. Book II. In this method the triangles should be chosen as nearly equilateral as possible; for if they be very oblique, the smallest error in the length of their sides will occasion a wide difference in the estimate of the surface. The calculation is much simpler from the application of Prop. 5. Book II. of the Elements, the base and altitude of each triangle only being measured; but that slovenly practice appears liable to great inaccuracy. The perpendicular may indeed be traced by help of the surveying cross, or more correctly by the box sextant, or the optical square, which is only the same instrument in a reduced and limited form; yet such repeated and unavoidable interruption to the progress of the work will probably more than counterbalance any advantage that might thence be gained. The usual mode of surveying a large estate, is to measure round it with the chain, and observe the angles at each turn by means of the theodolite. But these observations would require to be made with great care. If the boundaries of the estate be tolerably regular, it may be considered as a polygon, of which the angles, being necessarily very oblique, are therefore apt to affect the accuracy of the results. It would serve to rectify the conclusions, were such angles at each station conveniently divided, and the more distant signals observed. The best method of surveying, if not always the most expeditious, undoubtedly is to cover the ground with a series of connected triangles, planting the theodolite at each angular point, and computing from some base of considerable extent, which has been selected and measured with nice attention. The labour of transporting the instrument might also in many cases be abridged, by observing at any station the bearings at once of several signals. Angles can be measured more accurately than lines, and it might therefore be desirable that surveyors would generally employ theodolites of a better construction, and trust less to the aid of the chain. The quantity of surface marked out in this way is easily computed from trigonomet ry Adopting the general notation, the area of a triangle which has two sides, and their in ab cluded angle known, it is evident, will be denoted by 2*sinC, and the area of a triangle of which there are given all the a2 sin B sinC. 2 sin A angles and a side, is From the same principles may be determined the area of a quadrilateral figure inscribed in a circle. Let the sides a and b contain an acute angle A, and the opposite sides c and d must contain the obtuse supplémentary angle. The common base of these triangles, or diagonal of the quadrilateral figure, is hence |