5. The cosine of any angle of a plane triangle is equal to the sum of the squares of the adjacent sides minus the square of the side opposite, the whole divided by twice the product of the adjacent sides. 6. The area of any plane triangle is equal to one-half the product of any two sides into the sine of the included angle. 7. The area of any plane triangle is equal to the square root of the continued product of one-half its perimeter into one-half its perimeter minus each side separately. If the perimeter, a+b+c=p, then, b-c ·b+c 1. Having given two sides and the included angle, to find the other side and remaining angles.-Let b=30, c=20, and A=38° 20′; required a, B, and C. Find the angle B from the third formula of the fourth relation, which may be transposed to tan (B—C)=tan }(B+C)×; In this B+C=180°- A =180°-38° 20' 141° 40', and (B+C)=70° 50'; B-C is unknown; b+c =30+20=50, and b-c=30-20=10. By substitution, tan (B-C)=2.87700 X8.57540. From this (B−C)=29° 55′ (very nearly), and B-C=29° 55' X2=59° 50'. From this A+B=38° 20′+100° 45′=139° 5′, and C=180°- (A+B) = 180° -139° 5' 40° 55'. Find the side a from the first formula of the first relation, which may be transposed to read, a=bX sin A =30X sin 38° 20' =30X .62024 = 18.94. Re.98245 member that sin 100° 45′ = sin (180°-100° 45') = sin 79° 15′. 2. Having given two sides and the angle opposite one of them, to find the other side and remaining angles. Let the given parts of the triangle shown in Fig. 1, be A=38° 20′, b=30, and a= 18.94; from which it is required to find c, B, and C. Find the angle B from the first formula of the first relation, which transb 30.00 posed is sin B=-X sin A= 18.94 B=79° 15′ or 100° 45'. a Xsin 38° 20' : = 30.00 X.62024.98245; whence Unless the shape of the triangle is actually known it is impossible to tell which of these values of B should be taken. In fact, both of them are correct, as a study of the accompanying figure will show. As only A, b, and a, are fixed, it is apparent that a may occupy either position CB or CB and yet have the same value, 18.94. Such being the case, the angle at B may be (for the position CB=a' = 18.94) CBA =79° 15', or (for the position CB'=a= 18.94) CB'A = 100° 15'. Hence, angle C=180°- (A+B) = 180° (38° 20'+79° 15′)=180°-117° 35′ = 62° 25', or C=180°- (38° 20'+100° 45') = 180°-139° 5′ = 40° 55′ The side c may now be found from the second formula of the first relation, which may be transposed to read c=a C с B' FIG. 1 B angle C, c=18.94X sin 62° 25' = 18.94X sin C and taking the two values of the sin 38° 20' .62024 Thus two solutions of this triangle are possible; in the first case, B=79° 15', C=62° 25', c=27.07, and in the second case, B=100° 45', C=40° 55', c=20. 3. Having given two angles and any side, to find the other angle and the other two sides.-Let A=38° 20', B=100° 45', a=18.94; to find the remaining angle C and the other sides b and c. Find C from the relation C-180°- (A+B) = 180°— (38° 20'+100° 45') =180°-139° 5' -40° 55'. The sides may now be found from the first and second formulas given in the first relation after these have been transposed. b=a =18.94Xsin 38° 20′ c=a sin B sin A sin C sin A sin 100° 45' .98245 = 18.94 X =30 .62024 sin 40° 55' .65496 = sin 38° 20' 18.94 X =20 .62024 three angles.-Let a = 18.94, Using the formulas given in 4. Having given the three sides to find the b=30, and c=20; required the angles A, B, and C. the fifth relation to find A and B, then C may be found from C=180°- (A+B). b2c2-a2 302+202-18.942 C=180° (A+B) = 180°- (38° 20'+100° 45') = 40° 55' Note that the angle corresponding to the cosine .18648 is either 79° 15' or 100° 45'. By referring to the section Signs of Trigonometric Functions, it will be seen that when the cosine is minus, as it is in this case (-.18648), the angle is between 90° and 270°; hence, the value B=100° 45′ is taken. This example is readily solved by the solution of two Let fall a right-angled triangles, as shown in Fig. 2. perpendicular CD from the opposite vertex C upon the longest side AB dividing it into two segments AD =m and DB=n. From geometry, m+nb+a=b-a (b+a) (b−a) :m-n, and as m +n=c, m—n=· bining the value of m-n thus obtained with that of m+n=c, the values of m and n may be found. In the right-angled triangles ACD and BCD, b and m and a and n, respectively, are given, from which the angles A and B may be calculated; angle C found by subtracting the sum of angles A and B from 180°. C m FIG. 2 Using the values for the sides, a 18.94, b=30, and c=27.066 (30+18.94) X (30-18.94) 48.94 X 11.06 m-n= D n =19.998. Then =.18659. Whence B=79° 15'. C=180°- (A+B) = 180°-117° 35′ = 62° 25' Tables of natural and logarithmic trigonometric functions will be found at the end of the volume; each table is preceded by the necessary explanations for its use. m 23.532 =.78440. Whence A=38° 20'. SURVEYING THE COMPASS GENERAL DESCRIPTION Surveying is an extension of mensuration, and, as ordinarily practice may be divided into surface work, or ordinary surveying, and undergroun work, or mine surveying. With slight modifications, the instruments employe in both are the same, and consist of a compass-if the work is of little impo tance, and accuracy is not required-a transit, level, transit and level rod steel tape or chain, and measuring pins, and sometimes certain accessory instru ments, as clinometers or slope levels, dipping needles, etc., as will be describe later. The compass may be either a pocket compass, or a surveyor's compass and may be used while held in the hand, or upon a tripod. The Jacob's staf convenient for use on the surface, is useless in the mine. As the compass can not be sighted accurately on an object, cannot be read closer than 30', excep by guess, and may be deflected from its true course as much as 2° or 3° by th iron in the rails or water pipes or by electric currents, it is obvious that bear ings and angles determined through its use cannot be relied on as being withi 15 of the truth and they may be very much more in error. As present day surveying requires that any angle be known within 1 min. and in special cases such as tunnel work, within 30" or even 20" or 15", the compass is now no longer used except, in emergencies, when a transit is not available. However in driving room necks far enough for the permanent sights, in obtaining a rough idea of the direction of a heading, and, on the surface, in connection with the rerunning of old land lines, the compass has its uses. Owing to the length of time taken by the needle to settle so that it can be read, an accurate transit survey can commonly be made in less time than an inaccurate one with the compass. COMPASS ADJUSTMENTS When adjusting the levels, first bring the bubbles into the center by the pressure of the hand on different parts of the plate, and then turn the compass half way around. Should the bubbles run to the ends of the tubes, those ends are the higher; these should then be lowered by tightening the screws immediately under, and loosening those under the lower ends until, by estimation, the error is half removed. The plate should again be leveled and the first operation repeated until the bubbles will remain in the center during an entire revolution of the compass. The sights may next be tested by observing, through the slits, a fine hair or thread, made exactly vertical by a plumb. Should the hair appear on one side of the slit, the sight must be adjusted by filing off its under surface on the side that seems the higher. The needle is adjusted in the following manner: Having the eye nearly in the same plane with the graduated rim of the compass circle, with a small splinter of wood, or a slender iron wire, bring one end of the needle in line with any prominent division of the circle, as the 0 or 90° mark, and notice if the other end corresponds with the degree on the opposite side. If it does, the needle is said to cut opposite degrees; if not, bend the center pin by applying a small brass wrench, furnished with most compasses, about in. below the point of the pin, until the ends of the needle are brought into line with the opposite degrees. Then, holding the needle in the same position, turn the compass half way around, and note whether it now cuts opposite degrees; if not, correct half the error by bending the needle, and the remainder by bending the center pin. The operation must be repeated until perfect reversion is secured in the first position. This being obtained, it may be tried on another quarter of the circle; if any error is there manifested, the correction must be made in the center pin only, the needle being already straightened by the previous operation. When again made to cut, it should be tried on the other quarters of the circle, and corrections made in the same manner until the error is entirely removed, and the needle will reverse in every point of the divided circle. USING THE COMPASS When using the compass, the surveyor should keep the south end toward his person, and read the bearings from the north end of the needle. In the surveyor's compass the position of the E and W letters on the face of the compass are reversed from their natural position, in order that the direction of the sight may be correctly read. The compass circle being graduated to 1°, a little practice will enable the surveyor to read the bearings to quarters-estimating with his eye the space bisected by the point of the needle. The compass is divided into quadrants, and 0 is placed at the north and south ends; 90° is placed at the E and W marks, and the graduations run right and left from the 0 to 90°. When reading the bearing, the surveyor will notice that if the sights are pointed in a NW direction, the north end of the needle, which always points approximately north, is to the right of the front sight or front end of the telescope, and, as the number of degrees is read from it, the letters marking the cardinal points of the compass read correctly. If the E, or east, mark were on the right side of the circle, a NW course would read NE. This same remark applies to all four quadrants. The compass should always be in a level position. If all the corners of a field can be seen from a central point, the survey can be made by setting up at that point, and with one corner as a backsight, taking all the other corners as foresights, and by measuring from this point to all of the corners; or the compass can be set up at any corner and a line of survey run around the field. This latter method is called meandering. Both methods will give the same result when plotted; but the first is much quicker, as the boundaries of a tract are frequently overgrown with bushes that must be cleared to allow a sight; while a central point can frequently be found that will allow a free sight to all the corners, and the distance can be measured by tape, or stadia. As the central point is nearer the corners than they are to one another, a shorter distance must be chained or cut in the case of a central set-up. MAGNETIC VARIATION It Magnetic declination, or variation, of the needle is the angle made by the magnetic meridian with the true meridian or true north and south line. is east or west according as the north end of the needle lies east or west of the true meridian. It is not constant, but changes from year to year, and, for this reason, in rerunning the lines of a tract of land, from field notes of some years' standing, the surveyor makes an allowance in the bearing of every line by means of a vernier. The declination, where a knowledge of it is necessary, should always be determined for the particular place and at the particular time where and when it is needed. Quite a number of the States in cooperation with the United States Coast and Geodetic Survey have established a true meridian by astronomical observations at each county seat. Information as to the location, etc., of the monuments marking the meridian may be obtained from the county surveyor, the recorder of deeds, or some one else in authority at the county court house. However, the variation thus obtained is only available for use a comparatively short distance either east or west of the county seat (assuming that the highest accuracy is desired), because on the average, there is in the United States, a change in the value of the declination of 1' per mi. in the foregoing directions. From this, it is apparent that the declination at a place 30 mi. east or west of the county seat will probably vary 30' from that at the monuments referred to. This difference of 30' is within the limits between which the compass is ordinarily read. In proceeding north or south from the county seat, the change in declination is very much less than in an east or west direction. If the declination cannot be determined, a note should be made of the date of the survey, with a statement to the effect that the bearings are referred to the magnetic meridian, and these notes should appear on the map and should be incorporated in the deed if the survey was made preliminary to a transfer of property. The United States Coast and Geodetic Survey, Washington, District of Columbia, issues from time to time tables and charts showing the declination at many points in the United States and outlying possessions, together with formulas by means of which the declination may be calculated with a high degree of accuracy at future times. These may be obtained from the Superintendent of the Survey. Reading the Vernier.-The compass vernier, shown in the accompany illustration, is usually so graduated that 30 spaces on it equal 31 on the li of the instrument and, commonly, there are 15 spaces on each side of the 0 ma It is read as follows: Note the degrees and half degrees on the limb of instrument. If the space pas beyond the degree or half-deg mark by the zero mark on vernier is less than one-half 1 space of 1° on the limb, the nu ber of minutes is, of course, 1 than 15, and must be read from t lower row of figures. If the spa passed is greater than one-half t 20 25 15 10 30 25 20 0 1 spacing on the limb, the upper row of figures must be read. The line on t vernier that exactly coincides with a line on the limb is the mark that denot the number of minutes. If the index is moved to the right, the minutes are re from the left half of the vernier; if moved to the left, they are read from t right side of the vernier. Turning Off the Variation. Moving the vernier to either side, and with of course, the compass circle attached, set the compass to any variation placing the instrument on some well-defined line of the old survey, and turning the tangent screw (slow-motion screw) until the needle of the compa indicates the same bearing as that given in the old field notes of the origin survey. Then screw up the clamping nut underneath the vernier and run a the other lines from the old field notes without further alteration. The rea ing of the vernier on the limb gives the amount of variation since the origin survey was made. FIELD NOTES FOR AN OUTSIDE COMPASS SURVEY Call place of beginning Station 1. Stations Bearings At 1+ 37 ft. crossed small stream 3 ft. wide. At 1+116 ft. = first side of road. At 1+131 ft. second side of road. Distances 270.0 At 1+137 ft. blazed and painted pine tree, 3 ft. left, marked for a go-by Station 2 is a stake at foot of white-oak tree, blazed and painted on fou sides for corner. 2-3 Station 3 is a stake-and-stones corner. N 831° E 129.0 3-4 3+64 ft. 3+196 ft. S 57° E 222.0 center of small stream 2 ft. wide. 4-5 Station 4, cut stone corner. 4+174 ft. ledge of sandstone 10 ft. thick, dipping 27° south. 5+274 ft. ledge of sandstone 10 ft. thick, dipping 25° south (evidently continuation of same ledge as at 4+ 174). Station 1 place of beginning. THE TRANSIT GENERAL DESCRIPTION The transit is the only instrument that should be used for measuring angles in any survey where accuracy is desired. The advantages of a transit over a vernier compass are mainly due to the use of a telescope. By its use, angles can be measured either vertically or horizontally, and, as the vernier is used throughout, extreme accuracy is secured. Fig. 1 shows the interior construction of the sockets of a transit having two verniers to the limb, the manner in which it is detached from its spindle, and how it can be taken apart when desired. The limb b is attached to the main socket c, which is carefully fitted to the conical spindle h, and held in place by the spring catch s. The upper plate a, carrying the compass circle, standards, etc., is fastened to the flanges of the socket k, which is fitted to the upper conical surface of the |