FIELD NOTES OF THE EXTERIOR LINES OF AN ISOLATED TOWNSHIP. Field notes of the Survey of township 25 north, of range 2 west, of the Willamette meridian, in the Territory of OREGON, by Robert Acres, deputy surveyor, under his contruct No. 1, bearing date the 2d day of January, 1851. TOWNSHIP LINES commenced January 20, 1851. On a random line on the south boundaries of sections 31, 32, Therefore the correction will be 5 chains 47 links W. 37.1 Feet. West 40.00 TRUE SOUTHERN BOUNDARY variation 18° 41' E. a 10 62.50 a beech 24 in. dia. bears N. 11 E. 38 lks. dist. Land level, part wet and swampy; timber beech, oak, ash, Mound section corner. Deficient timbered corners. hickory, &c. 65.00 To beginning of hill 80.00 a 10 a 5 Set post, with trench, cor. of secs. 34 & 35, 2 & 3, from which a 20. a beech 10 in. dia. bears S. 51 E. 13 1. dist. Land level, rich, and good for farming; timber same. West. On the S. boundary of sec. 34— 40.00 Set qr. sec. post, with trench, from which a B. oak 10 in. dia. bears N. 2 E. 635 1. dist. Planted SW. a beech nut. a 5 80.00 To corner of sections 33, 34, 3 and 4, drove charred stakes a 10 Planted NE. a W. oak ac'n; NW. a yel. locust seed. Land level, rich and good for farming, some scattering oak &c., &c., &c. Land level, second rate; timber beech, poplar, sugar, and und'gr. spice, &c. Intersect E. boundary at post.. d 10 d 10 a 60 d 40 d 20 a 10 Random. Land level, second rate; timber, beech, oak, ash, &c. MEANDERS OF CHICKEELES RIVER. Beginning at a meander post in the northern township boundary, and thonce on the left bank down stream. Commenced February 11, 1851. S. 37 W. REMARKS. In section 4 bearing to corner sec. 4 on right bank N. 70° W. To post in line between sections 4 and 5, breadth of river by In section 5. S. 44 W. S. 36 W. 16.50 To upper corner of John Smith's claim, course E. 21.96 27.53 To post in line between sections 5 and 8, breadth of river by triangulation 8 chains 78 links. APPENDIX. APPENDIX A. SYNOPSIS OF PLANE TRIGONOMETRY.* (1) Definition. Plane Trigonometry is that branch of Mathematical Science which treats of the relations between the sides and angles of plane triangles. It teaches how to find any three of these six parts, when the other three are given and one of them, at least, is a side. (2) Angles and Arcs. The angles of a triangle are measured by the arcs described, with any radius, from the angular points as centres, and intercepted between the legs of the angles. These arcs are measured by comparing them with an entire circumference, described with the same radius. Every circumference is regarded as being divided into 360 equal parts, called degrees. Each degree is divided into 60 equal parts, called minutes, and each minute into 60 seconds. These divisions are indicated by the marks". Thus 28 degrees, 17 minutes, and 49 seconds, are written 28° 17′ 49′′. Fractions of a second are best expressed decimally. An arc, including a quarter of a circumference and measuring a right angle, is therefore 90°. A semicircumference comprises 180°. It is often represented by π, which equals 3.14159, &c., or 34 approximately, the radius being unity. The length of 1o in parts of radius = 0.01745329; that of 1'=0.00029089; and that of 10.00000485. The length of the radius of a circle in degrees, or 360ths of the circumference = 57°.29578 — 57° 17′ 24′′.8=3437′.747 = 206264′′.8.† An arc may be regarded as generated by a point, M, moving from an origin, A, around a circle, in the direction of the arrow. The point may thus describe arcs of any lengths, such as AM; AB = 90° = π; ABC: π; ABC= 180°: ABCD = 270° =3π; ABCDA= 360° = 2 π. 2 T The point may still continue its motion, and generate arcs greater than a circumference, or than two circumferences, or than three; or even infinite in length. While the point, M, describes these arcs, the radius, OM, indefinitely produced, generates corresponding angles. Fig. 397. M C A 0 D For merely solving triangles, only Articles (1), (2), (3), (5), (6), (10), (11), and (12), are needed. + The number of seconds in any arc which is given in parts of radius, radius being unity, equals the length of the arc so given divided by the length of the arc of one second; or multiplied by the number of seconds in radius. If the point, M, should move from the origin, A, in the contrary direction to its former movement, the arcs generated by it are regarded as negative, or minus; and so too, of necessity, the angles measured by the arcs. Arcs and angles may therefore vary in length from 0 to +∞ in one direction, and from 0 to - ∞ in the contrary direction. The Complement of an arc is the arc which would remain after subtracting the arc from a quarter of the circumference, or from 90°. If the arc be more than 90°, its complement is necessarily negative. The Supplement of an are is what would remain after subtracting it from half the circumference, or from 180°. If the arc be more than 180°, its supplement is necessarily negative. (3) Trigonometrical Lines. The relations of the sides of a triangle to its angles are what is required; but it is more convenient to replace the angles by arcs; and, once more, to replace the arcs by certain straight lines depending upon them, and increasing and decreasing with them, or conversely, in such a way that the length of the lines can be found from that of the arcs, and vice versa. It is with these lines that the sides of a triangle are compared.* These lines are called Trigonometrical Lines; or Circular Functions, because their length is a function of that of the circular arcs. The principal Trigonometrical lines are Sines, Tangents, and Secants. Chords and versed sines are also used. The SINE of an arc, AM, is the perpendicular, MP, let fall, from one extremity of the arc, upon the diameter which passes through the other extremity. The TANGENT of an arc, AM, is the distance, AT, intercepted, on the tangent drawn at one extremity of the arc, between that extremity and the prolongation of the radius which passes through the other extremity. The SECANT of an arc, AM, is the part, OT, of the prolonged radius, comprised between the centre and the tangent. Fig. 398. T M A P The sine, tangent, and secant of the complement of an arc are called the CoSINE, Co-Tangent, and CO-SECANT of that arc. Thus, MQ is the cosine of AM, BS its cotangent, and OS its cosecant. The cosine MQ is equal to OP, the part of the radius comprised between the centre and the foot of the sine. The chord of an arc is equal to twice the sine of half that arc. The versed-sine of an arc, AM, is the distance, AP, comprised between the origin of the arc and the foot of the sine. It is consequently equal to the difference between the radius and the sine. The Trigonometrical lines are usually written in an abbreviated form. Calling the arc AMa, we write, MP sin. a. MQ= cos. α. AT= =tan. a. OT sec. a. The period after sin., tan., &c., indicating abbreviation, is frequently omitted. The arcs whose sines, tangents, &c., are equal to a line =a, are written, * For the great value of this indirect mode of comparing the sides and angles of triangles, see Comte's "Philosophy of Mathematics," (Harpers', 1851,) page 225. When the radius of the arcs which measure the angles is unity, these ratios may be used for the lines. If the radius be any other length, the results which have been obtained by the above supposition, must be modified by dividing each of the trigonometrical lines in the result by radius, and thus rendering the equations of the results "homogeneous." The same effect would be produced by multiplying each term in the expression by such a power of radius as would make it contain a number of linear factors equal to the greatest number in any term. The radius is usually represented by r, or R. (5) Their variations in length. As the point M moves around the circle, and the arc thus increases, the sines, tangents, and secants, starting from zero, also increase; till, when the Fig. 400. M G Ί A P point M has arrived at B, and the arc has become 90°, the sine has become equal to S radius, or unity, and the tangent and secant have become infinite. The complementary lines have decreased; the cosine being equal to radius or unity at starting and becoming zero, and the cotangent and cosecant passing from infinity to zero. When the point M has passed the first quadrant at B and is proceeding towards C, the sines, tangents, and secants begin to decrease, till, when the point has reached C, they have the same values as at A. They then begin to increase again, and so on. The Table on page 382 indicates these variations. N D N T The sines and tangents of very small arcs may be regarded as sensibly proportional to the arcs themselves; so that for sin. a", we may write a . sin. 1"; and similarly, though less accurately, for sin. a', we may write a. sin. 1'. The sines and tangents of very small ares may similarly be regarded as sensibly of the same length as the arcs themselves.* * Consequently, the note on page 379 may read thus: The number of seconds in any very small arc given in parts of radius, radius being unity, is equal to the length of the arc so given divided by sin. 1". |