MISCELLANEOUS. When a survey is calculated by chains and links, and the contents stand in acres and decimal parts of an acre, it may be multiplied by the price of an acre, and the product will be the amount. EXAMPLE. A piece of land, 12 chains and 25 links in length, and 10 chains and 25 links in breadth, is sold for $20 25, per acre ; -what is the price of it ? Length 12.25 6125 2450 1225 Acres and decimal parts 12.55625 Price of an acre 20.25 6278125 2511250 2511250 Answer $254,26.40625 The writer knows not who invented the following rules for finding contained angles. For plainness, they are not exceeded. * When the first letters are alike, and the two N. 62° E. N. 44 W. last are unlike, add the degrees of both courses together, which gives the contained angle. When the two first and the two last letters are S. 72o E. alike, subtract one course from the other, and S. 25 E. the remainder will be the contained angle. When the two first letters are unlike, and the N. 64o E. two last alike, add both courses together, and S. 35 E. subtract their sum from 180, the remainder will be the contained angle. When the two first and the two last letters N. 57° W. are unlike, subtract one course from the other, S. 25 E. the remainder from 180, and the remainder will be the contained angle. * These rules are nothing different in principle, from the common rules, page 50, and, in their practical application, by no means as simple. To one who understands the theory of the common rules, they are the most simple, which can be invented ; and to one who does not, APPLICATION OF THE ABOVE RULES. Two courses are given, viz. N. 679 W. and 28° E. to find the angle.-Suppose yourself standing at the point where these courses meet. Reverse the letters of the first course, and they will stand thus, S. 67° E.) The third rule applies in N. 28 E. this case. When the quantity of any angle in a survey is wanted, the preceding course must be reversed; then both courses will run from the same point. versing of the letters of the first course is troublesome and perplexing. Perhaps, however, for practical men, the common rules may be improve ed in some degree, by expressing them as follows. I. IF THE FIRST LETTERS ARE ALIKE, AND THE LAST ALSO, ADD THE LESS COURSE TO THE SUPPLEMENT OF THE GREATER. II. IF THE FIRST LETTERS ARE ALIKE, AND THE LAST UNLIKE, SUBTRACT THE SUM OF THE COURSES FROM 180°. III. IF THE FIRST LETTERS ARE UNLIKE, AND THE LAST ALIKE, ADD THE COURSES. IV. IF THE FIRST LETTERS ARE UNLIKE, AND THE LAST ALSO, SUBTRACT THE LESS COURSE FROM THE GREATER. In employing these rules, the letters may be taken as they stand in the FIELD BOOK. A set of figures like the following well fixed in the mind, will be a great assistance to the learner. RULE I. RULE II. CONVERGING OF MERIDIANS. The length of a degree of longitude in any parallel of lati. tude is to the length of a degree upon the equator, as the co-sine of that latitude is to radius. R. : 60 miles* : : co-sine of the lat. : the length of a degree on that lat. It will be seen that the required angle, in each case, is ABC. A set of figures of this kind, once fully understood, and fixed in the memory, will be the best means of rendering this subject simple. We recom. mend to all learners, to study the above with attention.-ED. * This proportion is for geographical miles or minutes of a great circle of the earth. Of course, to the practical man, it is of little use. For statute miles we should employ, instead of 60, the number of statute miles in a degree. This number has been commonly laid down at 69}. This estimate is, however, undoubtedly too great. It is, notwithstanding, difficult to assign the true length of a degree, when our measures themselves are liable to change. Until some universal and permanent standard of length shall be adopted throughout the country, we shall hardly be able to speak with much certainty on this subject. The ten millionth of a quadrant of a meridian of the earth has been adopted for a standard in France. In England, and of late years in the state New York, the pendulum has been made the regulator of measures of length. In modern English measure, a degree is but little more than 692' miles, acccording to the measurements most relied on. When it is recollected that between the various measurements, which have been made, of degrees on the earth's surface, there have been remarkable differences ; that in different parts of the world, the degrees vary in length, increasing always from the equator to the poles ; and that a very trifling variation in smaller measures, as a foot, or yard, may become very perceptible in a distance of 60 or 70 miles, the diffi TO CHANGE A MAP FROM ONE SCALE TO ANOTHER.* Fig. 4. D A Suppose the map to be ABCD. It is desired to draw a larger map of the same field, without the necessity of plotting a. gain. Take any point within the field, as E. From this point draw lines E of indefinite length thro'. B В a the ratio, in which you wish to increase the size of your map. Through a draw ab, parallel to AB. Through b, the point where ab intersects EB extended, draw bc, parallel to BC. Through c, the point where be intersects EC extended, draw cd parallel to CD. Through d, the point where cd in. tersects CD extended, draw da, parallel to DA. The line da will intersect EA extended, at the starting point, and abcd will be the enlarged map. If it be required to draw a map smaller than the given one, the point a may be taken within the field, and the process will be similar. If the plot, which it is desired to enlarge, be near one side of the sheet, so that it cannot be extended in that direction, At present, it is most generally considered about 69] miles. Considering it as such, the proportion above, will become, Re: 69} miles :: Co-sin. lat. : length of deg. in that lat. If, then, the latitude be 60°, as above, the length of a a degree will be 34 miles.-Ed. * This and the following problems are not contained in previous edi. tions.-ED. the point E may be taken in one side or angle. And generally, it may be observed, that the nearer the point E is taken to one side of the map, the less will the enlarged plot ex. tend in that direction. Thus, in the figure, E is nearest the side AB, and the map extends least on that side. TO FIND THE AREA OF A FIELD GEOMETRICALLY, BY REDUCING IT TO A TRIANGLE. Fig. 5. K B In many cases it is convenient to be able to make an estimate of the quantity of land in a field, with tolerable accuracy, but without the trouble of a tedious computation. In such instances, the plot may often be reduced, with advantage, to a triangle. Let ABCDEF be the field. I observe a re-entering angle, on the north side, viz. CDE. If Ijoin CE, the plot will contain too much, by the triangle CDE. If I, then, cut off a portion, equal to CDE, the proper quantity will be left. I draw DG, parallel to EC, and join EG. Since DG is parallel to EC, the height of the triangle EGC, is equal to the height of the triangle CDE. And as they stand on the same base, EC, EGČ is equal to CDE; for the area of each is found by multiplying half the base into the height. Taking away EGC, then, there is left the figure ABGEF, equal to the ori. ginal plot, but having one side less. Thus the side EG has been substituted for the two sides, ED, DC. In like manner, by joining EB, and drawing a line from G to H, parallel to EB, I may substitute EH for the two sides |