LIBRAR UNIVERSITY OF CALIFORNIA Received Accession No . Clazs No. . INSTRUCTIONS FOR USING DOLMAN'S ...NEW... DECIMAL, SCALE MEASURE PROTRACTOR. ~"~"~^"lfe$!S (TJHIVSRSITT ESSS I7M55) INSTRUCTIONS FOR USING DOLMAN'S ...NEW... DECIMAL SCALE MEASURE PROTRACTOR. This protractor produces length and position of all lines and degrees of angles required by algebra and calculus by practical object lessons in scale measure, thereby solving millions of problems by measurement in construc- tive geometry without algebra or calculus. DOLMAN'S PROTRACTOR. DESCRIPTION AND USE OP DOLMAN'S NEW DECIMAL SCALE MEASURE PRO- TRACTOR. This protractor is a semi- circle with the semi-circumference of the pro- tractor graduated 90 degrees from OM line to the right, and left to the bottom line DD of the protractor. The OM, or meridian line, meets bottom line, DD, at the middle, and in the centra of the small semi-circle and at right angles to line DD. The two arms, BB, BB, must be fastened by a needle at the intersection of lines DD and O. M. in the small S3mi-circle. A parallelogram two by four inches, is cut out of the middle of the protractor parallel to sides LL, LL and DD, DD, and from equi-angular parallels, and by revolving the pro tractor one-half around, using the nesdle as a pivot, a four inch square, and a circle can be formed with a common centre to square and circle. The movable bar, DD, moves parallel to bottom line, DD. An elastic scale, seven inches long, is for measuring arcs, or any other measurement of lines. The inside edges are graduated to twenty spaces to one inch, and sidss LL LL read from one at bottom line, DD, up to 200 on out- side lines, and 40 on inside lines. Bottom and top lines, DD DD, read right and left from OM line 200 on outside lines and 40 on inside lines. Arms BB BB read from center needle outward 350 on outside lines and 70 on inside lines. All in edges are graduated to twenty spaces to one inch. The graduations of degrees on the semi-circle are not units of length. Their departure of length of arc depends on the length of the sides and number of degrees of the angle. USE OF THE DECIMAL SCALE MEASURE PROTRACTOR. This protractor measures degrees of angles and gives the length of every line to scale measure, and is a guide to draw every line by, without calculat- ing the length of any lines. No other instrument in use at this time gives degrees of angles, length of lines and a guide to draw the lines of every polygon. Decimal is a scale of which the order of progression uniformly is ten. A scale is a system of measurement that small spaces are used to represent larger units of measure and greater numbers of units proportionally, viz : To represent large area on small space, as maps, charts, plots, diagrams, &c. A decimal scale progresses thus: If one-tenth of one inch represents one unit, one inch in length would represent ten units of length, and one square inch would represent one hundred squares of one-tenth of squares of one square inch, and ten square inches, or 3 16-100 inches square would represent one thousand one -tenths of inches in two dimensions. If we assume that one-tenth of one inch of scale in one dimension shall represent $1,000, then Vne inch in one dimension would represent $10,000, and one square inch in two dimensions would represent $100,000, and ten square inches, or 3 16-100 square would represent 81,090,090, and ons hundred square inches, or ten inches square would represent '$10,000,000, and one thousand square inches, or DOLMAN'S PROTRACTOR. 31 62-100 inches square would represent 8100,000,000, and ten thousand square inches, or one hundred inches square, would represent by scale measure, 81,- 030,000,000. The above explanation of decimal scale measure as applied to quantity, by numbers of units compared with extension of lines and angles are given to assist the mind to comprehend quantity and magnitude as multiplied by 10, 100, 1,000 &c. Arithmetic is the science of numbers applied to units of quantity. Geometry is the science of measurement. Measurement is, first, ascertaining the number of units in a line by com- parison in extension in one dimension called distance ; second, by comparing square units with area in two dimensions called square measure; third, by comparing square units with thickness extension in three . dimensions called cubic measure. Algebra is the science of ascertaining unknown numbers of units of quan- tity by subtracting one known number of units from other known numbers of units, or by adding, multiplying or dividing or all combined.* The following tables of units are in common use in the United States, and are in the arithmetic, viz: First, units of length, as inches, feet, miles, &c. Second, units of area, or square units, as square inches, square feet, square acres, &c. Third, units of volume, or cubic measure, as cubic inches, cubic feet, &c. Fourth, units of angles 360 degrees in every circle, and each degree may be considered an angle; 21,600 minutes in every circle, and each minute may constitute a separate angle. Every circle is divisible into (1,296,000) one million two hundred and ninety- six thousand angles of one second each, or any number less than the angular space may be an angle. Fifth, units of gravity (weight) determined by comparing the volume of water as a standard of 287-s cubic inches of water equals one pound avoirdupois weight and also equals one pint of liquid measure. The Winchester Bushel contains 2150 42-100 cubic inches, or 77627-1000 pounds of water avoirdupois, or 5760 grains apothecary weight. One ounce of Troy equals 480 grains, and also equals 437}^ grains avoirdu- pois weight. Sixth, units of duration of time determined by motion of the planets, and are units of seconds, minutes, hours, days and years. Seventh, units of value are created by law, and may be cents and dollars, shillings and pence, or any other unit desired. Degrees of angles have no proportion of length as to scale measure of departure of the arc of any angle. The right angled triangle is a unit of comparison of degrees of angles, and measured departure of angles. Every right angled triangle is the one -half of a square, provided the two short sides of the triangle are of equal length, and one angle of the triangle will contain 90 degrees, if the short sides of the right angled triangle are of dif- ferent length, the triangle is the one-half of an equiangular parallelogram. A right angle has two sides and always contains 90 degrees, no more, no *llie remainder ttitvr subtraction is the unknown <|iiantity. DOLMAN'S PROTRACTOR. less. Every right angled triangle has three sides and three angles, and one of the angles is always equal to 90 degrees, and the other two angles are equal to 90 degrees. Every equilateral triangle has three sides of equal length, and three angles of 60 degrees each. Every scalene triangle has three sides of unequal length and three angles, and the three angles combined equal 180 degrees, and the scalene triangle can always be divided into two right angled triangles of unequal dimensions. Problems have but one demand that is how much and that demand is satisfied by adding quantity to quantity, or by subtracting quantity from quantity. All lines must have position, and that position and the relation to the position of other lines give names to the lines and cause the names of lines to change. The triangle, square and hexagon, are the only regular polygons by which the angular space about a point can be completely filled up. Quantity is a general term applied to every thing which can be increased, diminished, measured, compared, or estimated. It embraces numbers and magnitude. DEFINITION. RADII IS THE PLURAL OF RADIUS. A radius is any straight line passing from the centre of the circle to the circumference of the circle. Diameter is any straight line passing through the centre of a circle from one side of the circumference to the opposite side of the circle, or any polygon. Circle, or circumference, is a line which is equal distance from a point within, called centre. Perimeter, is any number of straight lines that enclose a polygon. A polygon is any diagram with three or more sides. A diagonal line is a straight line drawn from one angle to an opposite angle. A vertex is the point where two lines meet, that form angles. Base line, or meridian line, is the beginning line, or line that all other lines conform to. A perpendicular line is a line that meets another line at right angle, called departure. A hypothenuse line is the longest line of every right angled triangle, and forms the third side of the triangle, and in surveying land is called the bearing Hne. DOLMAN'S PROTRACTOR. Co-tangent, is a line that will meat the tangent at right angle and starts from the same circumference 90 degrees from where the tangent starts. Co-ordinate triangle, means another triangle of equal dimensions of the first triangle, and opposite to the first trian 416. Ratio, is that relation between two quantities which is expressed by the quotient of the first divided by the second. Thus the ratio of 4 to 12 is 12-4, or, 12 divided by 4, the ratio is 3. A proportion is an equality of ratios. Infinity in measurement is when two lines approach so near to each other that no perceivable difierence can be seen. Inscribed means one polygon, or circle produced within another circle or p ;>!ygon. Described means a circle or polygon produced around another circle or polygon. Applicati- n of the decimal scale measure protractor to the following five diaijrams, as per instructions, will enable the student to determine quantities and magnitude by measurement without algebra, or intricate calculus: DIAGRAM NO. 1. This diagram represents first a plane, second a sphere, third the propor- tion of area of a circle to the area of a square, which is, circle area, 7854 Area of square is, square .10000 The centre of this circle represents in geometry, first, the point of hegln- ning of a space. Second, the centre of a sphjie or the earth. The centre of DOLMAN'S PROTRACTOR. Square the two short sides, add their products and extract the square root of the product. This earth is, in geometry, conceived to be the centre of space, and all con- ceivable lines that start from the centre of this earth are considered in geom- etry as perpendicular lines, and all lines produced at right angles to the perpendicular lines are considered as horizontal lines. The lines that are conceived to pass from the csntre of the earth to its surface are represented by a plummet, and the horizontal lines are represented by the level, and are also called tangants, if continued in a straight line, and are known as apparent levels, or horizontal lines. The lines produced on the circumference of the earth by the level when taken at different points of the earth's surface, produce circular line s, called true level, and are always at right angles to the plumb lines caused by the change of position of the level on the circumference of the earth as the Iev3l will be at right angle to every plumb line at every point of the earth's surface and the plumb line changes toward the centre of the earth when moved from one point to any other point on the earth's surface. Apparent level, or tan- gent lines only change four times, until they meet and form a square whoso sides are equal in length to the length of the diameter of the inscribed circle. The centre of diagram No. 1, as a plane, represents the beginning point of measurement by lines and angles. TO ADJUST THE PROTRACTOR FOR PLOTTING AND MEASURING LINES AND ANGLES. Insert a needle or pin into the holes in the centre of the half circles on BB, BB, with the graduated edges of the arms right and left, then insert the needle into the hole in the half circle on bottom line DD, place movable bar, DD, inside the square with the graduated edges parallel to bottom line, DP, then insert the point of the needle into the paper, and proceed to construct lines and angles according to requirements of the parts of diagram as require d by terms of the problem. Meridian, or latitude line, is a line produced, or conceived north and south. A longitude, or departure line, is a line produced east or west, and departs from a latitude, or base line, at right angle. Sine of arc, is the number of degrees and length of the perpendicular line to a radius of a circle. Co -sine, is the degrees and length of the adjoining side of the triangle of which the rxdius is a parallel, and the perpendicular is the sine. Secant, is the hypothenuse of the right angled triangle of the radius and tangent. Co-secant, is the hypothenuss of the complement angle.* Tangent line, is a line that is produced at right angle to a radius, and if the secant is at an angle of 45 degrees to the tangent line, the tangent and radius will be of equal length. Otherwise, the tangent will be longer than *W hen the degrees of an an tfe are subtracted from 90 degrees, the rpmaining degrees are called the complement; and when the degrees of an angle are subtracted from 180 degrees, the remainder is called the supplement angle. See co-sine, co-tangent etc., in diagram No 4 DOLMAN'S PROTRACTOR. the radius.* To trace the lines of diagram No. 1, with the protractor as an object lesson in drawing, adjust the protractor as instructed, then insert the needle into the centre of diagram No. 1, place the bottom line, DD, of the protractor on one of the diameters of the circle and movable bar, DD, will be parallel with the tengents and chords of the square. Sides LL, LL, will be parallel at righ* angles to lines DD, DD, and arms BB, BB, will be movable to coincide with radius or bearing lines. Movable bar DD, will indicate departure, tangents and chords, and sides LL, LL, can be moved to indicate latitude and BB, BB, will represent meridian lines. The OM line will always be at right angles to DD, DD, and parallel to LL, LL, and to mark opposite parallels must not be omitted when bearings change a igleS; and the protractor is to be moved to another bearing. The elastic scale will give the length of all lines and arcs to scale measure. A thorough knowledge of geometry can only be acquired by producing lines and angles to scale measure. To construct a circle, use a strip of card board for a radius, use a needle for a centre pivot, get the length of the radius from any part of the protrac- tor, except the degrees; make a small hole in the card board for a pencil point to mark the circumference of the circle and a needle for a centre. To form a square, use bottom line, DD, of the protractor cne side of the square, and one of side LL for the next side, then invert the protractor and use the same sides and same length of lines to construct the other sides of the square. To construct parallels, move bar DD the distance from bottom line DD that the parallels are required apart, and see that both ends of bar DD are the same distance from bottom line DD. To construct a right angled triangle, place one arm on the OM line of the protractor, place the other arm the given number of degrees from the first arm, move parallel bar DD to the given number on the arm that represents the given line, or if the given line is departure, move bar DD until the given number of graduations on bar DD fills the space between the two arms BB, BB and the two arms from bar DD to the needle will be the length of the other two sides of the right angled triangle. When the bearing of a right angled triangle is given, that is the number of degrees of departure from the meridian line, and the length of the bearing line given to obtain the length of the latitude line, and the length of the de- parture line of the right angled triangle, place one arm on the OM line of the protractor, and place the other arm the number of degrees to the right or left of the OM line, then move bar DD parallel until the number of graduations given is found on the arm that is not on the OM line, then the arm on the OM line between the bar and needle will be latitude, and the distance on bar DD between the two arms will be the length of departure, and the other arm will give bearing distance. To construct an equilateral triangle, place one arm 30 degrees to the right *Secant lines pass outside of the circle and meet the tangent line. BOLIVIAN'S PROTRACTOR. o? the OM line and the other arm 30 degrees to the left of the OM line; mov3 bar DD until the distance on bar DD is equal to the length on each arm. The i will the three sides be of equal length, and the three angles contain 60 degree 3 each. Move one arm to OM and it gives the perpendicular of the triangle. To construct a tangent square, or described square, to a circle, construct two diameters at right angles to the circle, divid ng the circle into four equal parts; place bottom line. DD and side LL on the outside of the circle, so that the length of the radius of the circle on DD and LL will meet two ends of the diameters will produce the first one -fourth of the square. Then move the protractor to the ends of the next diameters, and so on until the square is completed.* A line from the centre of the circle to the angle of the described square of a circle will be a radius that will double the arsa of the first circle, and the diagonal of the described square will be the length of a square double the area of the described square. RULES FOR CALCULATING LENGTH OF LINES, NUMBER OF SQUARE UNITS IN AREA AND CUBIC UNITS OF VOLUME. The diameter of a circle given, required the Icngtli of the circumference of the circle. RULE BY CALCULATION. Multiply the length of the diameter of the circle by 3.1416; point off four figures on the right of the product for decimals, and the remaining figures will be whole units. The circumference of a circle given, required the diam- eter of the circle. RULE BY CALCULATION. Add four ciphers to the circumference, then divide by 3.1416, and point off four right hand figures for decimals in the quotient. The diameter of a circle given, required the area of the circle. RULE BY CALCULATION. Multiply one-half of the diameter by one-half of the circumference, or square the diameter of the circle, and multiply that product by the decimal .7854. Point off four right hand f gures for decimals. The two short sides of a right angled triangle given, required tie area of the triangle. RULE BY CALCULATION. Multiply one short side by one-half the length of the other short side. The two short sides of a right angled triangle given, required the length of the bearing or hypothenuse. RULE BY CALCULATION. *>ce Di 'gram No. 1 DOLMAN'S PROTRACTOR. Sqaare the two short sides, add their products and extract the square root of the product. RULE BY PROTRACTOR. Piaca one arm of the protractor on the OM line, place the other arm on ths givaa number of degrees of departurs of the angle, move bar DD parallel until the number given for latitude on first arm, and the other arm from needle to bar DD will be the length of the bearing line. Ths diameter of a circle given, required the length of the side of the greatest inscribed square. RULE BY CALCULATION . Multiply the diameter by the decimal .7070, and cut off four decimals. RULE BY PROTRACTOR. Construct a circle, construct two diameters at 90 degrees, dividing the circle into four equal parts. The distance between any two ends of the diam- eters will be the length of the sides of the inscribed square. Straight lines drawn between the ends of the diameters will construct the inscribed square, and the four sides will be four chords to the four arcs made by the two diam- eters of the circle. The length of the side of a square given, required the area cf the square. RULE BY CALCULATION. Multiply the length of the side by its own length.* The area of a square given required the area of a circle, whose diameter is equal to the side of the square. RULE BY CALCULATION. Multiply the area of the square by the decimal .7854, and point off four decimals on the right. The diameter of a circle given, required the length of the side of a square whose area equals the area of the circle. RULE BY CALCULATION. Multiply the diameter of the circle by the decimal .8862, and cut off four decimals. The length of the side of an equilateral triangle given, required the per- pendicular of the triangle. RULE BY CALCULATION . Multiply the length of the side by the decimal .8660, and point off four decimals. RULE BY PROTRACTOR. Construct the triangle and measure the distance from any one of the angles to the center of the opposite side. To multiply a number by its own length is called squaring a number. 10 DOLMAN'S PROTRACTOR. The ci -cumf erence of a circle given, required the length of the side of a squa e equil in area to the area of the circle. RULE BY CALCULATION. Add three ciphers to the circumference, then divide by 4.442, and point off three decimals in the quotient. The diameter, or base, and length of a cylinder given, required the volume or cubic contents. RULE BY CALCULATION. Multiply the square of the base by the decimal .7854, and that product by the height of the cylinder, point off four decimals. The diameter and height of a cylinder given, required the superficial con- tents (area) of the cylinder. RULE BY CALCULATION. Mu biply the diameter of the cylinder by 3.1416, and multiply that product by tne height of the cylinder; point off four decicals, and that product will be the perpendicular superficial contents of the cylinder, less the two ends of the cylinder; multiply the squares of the diameter of the cylinder by two, and multiply that product by the decimal .7854; point off four decimals. The area of a sphere is equal to the area of four circles, whose diameters are equal to the diameter of the sphere. The diameter of a sphere given, required the area of the sphere. RULE BY CALCULATION. Square the diameter of the sphere, multiply that product by four, and multiply that product by the decimal .7854, and point off four decimals. The area and diameter of a sphere given, required the volume, or cubic contents of the sphere. RULE BY CALCULATION. Multiply the area of the sphere by one- sixth of the diameter of the sphere. The perpendicular of an equilateral triangle is three -fourths the length of the diameter of its described circle, and two-thirds the distance from the vertex to the opposite side on the perpendicular will be the centre of the described circle. Multiply the length of the side of an equilateral triangle by twenty and divide that product by twenty-three. This will very nearly give the length of the perpendicular of the triangle. RULE BY PROTRACTOR. Measure the perpendicular of the triangle with the protractor. The length of radius and degrees of arc given, required the length of the arc. RULE BY CALCULATION. Multiply the radius by 3.1416, divide that product by 180, and multiply that quotient by the number of degrees of the arc. This will givie the length DOLMAN'S PROTRACTOR. 11 of the arc. Point off four decimals in the last product.* Every square is divisible into two right angled triangles of equal dimen- sions and the two short sides of each triangle will be of equal length, and the diagonal line that separates the two right angled triangles will be the hypoth- enuse to both right angled triangles. If one short side of the right angled triangle is longer than the other short side of each right angled triangle, its polygon is an equiangular parallelogram. The side of a sector whose angle is 60 degrees given, required the length of the arc.f RULE BY CALCULATION. Multiply the length of the side of the sector by 3.1416; divide that product by three, and point off four decimals in the product. Co-ordinate angle means another angle equal to the first angle with opposite bearings. The decimal scale measure protractor does not give area and volume of quantities. This protractor gives length of lines and degrees of departure of angles, and the form of all angles and diagrams. All area and volume are ascertained by multiplying the length of a Jin 9 by its own length, called squaring a line, and multiplying the square product by the thickness gives volume, or cubes. Multiplying one long side by one short side gives area of equiangular par- allelograms DIAGRAM NO. 2. Rule by Protractor. Measure the arc with the elastic scale, by Protractor. -Measure the arc with the elastic scale. 12 DOLMAN'S PROTRACTOR. Diagram No. 2 is the bass of the principle of algebra. The length of the radius of the large circle of diagram No. 2 given, required the diagram by the protractor, and the area of each separate part of the diagram by simple calculation, viz : Addition, subtraction, multiplication and division. RULE FOR PROTRACTOR. Driw large circle, A, B, C, D, E, F, to given radius; divide large circle into six equal parts by constructing diameters, AD, BE, FC, 60 degrees in each division; place the centre of the protractor at D on large circle, and top of protractor, OM line, on A; move right arm 20 degrees to the right from OM line, and at the point that the arm crosses the second line, viz: BE win be the centre of first small circle; move left arm 20 degrees to the left of OM line, and where the left arm crosses the second line, FC, will be the centre of the second small circle; move centre of protractor to point B; move top of protractor, OM line, to E ; move left arm 20 degrees to left of OM line, and the point on second line, DA, will be the centre of third small circle. Move right arm 20 degrees to the right of OM line, and the right arm will meet the centre of first small circle. If the diagram is correctly constructed, the lines Y B, Y D, and Y F, are the radii of the three small circles, and all the straight lines of the diagram can be measured by the protractor to form the tangents or described square around the great circle. Place bottom line DD and side LL on the outside of large circle, placing side DD at either diameter on large circle and move the protractor until the length of the radius of the large circle meets the circle that will form the first one-fcurlh of the square, and the line DD will be first tangent line, and line LL will be first co- tangent line. Form other tangents the same way. CALCULATION OP THE AREA OF THE SEVERAL DIVISIONS OF PIAGKAM NO. 2 IY ARITHMETIC. First required is the area of large circle. RULE. Square the diameter of the large circle, multiplying that product by the decimal .7854, anjl point off four decimals. Second, estimate area of small circle by the same rule which applies to large circle. Third, ascertain area of equilateral triangle Y, Y, Y, by multiplying length of perpendicular by one-half the length of Y, Y, Y. Subtract one -half the area of one small circle from the area of triangle Y, Y, Y, and the remainder equal the six small divisions around the centre of large circle. Divide A ^ *? mainder bv 8ix > and Jt wil1 g^e the area of one of the small divisions. ie area of the three small circles and the area of the six small divisions, and subtract that product from the area of the large circle, which will give area of the .six large irregular divisions of the large circle, and divide that emamder by six, which will give the area of one of the large irregular divis- DOLMAN'S PROTRACTOR. 13 ions of the circle. Subtract the area of the large circle from the area of the described square. The remainder will equal the four irregular divisions of the square. That remainder divided by four will give the area of one of the four irregular divisions of the square. Careful inspection of instructions given for diagram No. 2 will show that the twelve irregular divisions of the large circle can not be measured. The only process to obtain their area is to subtract one known area from another known area, either by arithmetic or by algebra. The equilateral triangle, Y, Y, Y, separates the sector, or one- sixth part of the area of each of the three small circles; hence, three-sixths equal one- fcalf of the area of one small circle, and the remainder of the area of triangle Y, Y, Y, equals unknown quantities of the six small divisions, and the six large divisions and four divisions of the square are obtained by the same rule. DIAGRAM NO. 3. Diagram No. 3 is given to show the proportion and similarity of all right angled triangles and the use of latitude and departure and the principle of longitude sines, tangents, etc. To construct Diagram No. 3, place centre of protractor at A and OM line on D, and bottom line DD will be on A, B. Place right arm on C, 53 degrees from meridian line A, D , N: A, C, will be bearing north, and 53 degrees east. Let line A, C, be 50 in length to C. We want to know how far south it is from* C to B, and how far west it is from B to A. Move bar DD parallel to C> and line A, D, will indicate 30 latitude north on side LL and on the arm placed *].alituile sanure \v 11 DOLMAN'S PROTRACTOR. on OM line on line A, D, N, and bar DD will indicate 40 east from D to C. Move the protractor centre to C, place line DD of protractor on D of diagram and OM line en C, B. Move bar DD parallel to A, B; then will C, B indicate south latitude from C to B, and bar DD will indicate west departure 40 from B to A. The solid line, A, C; C, B; B, A, may represent a right angled trian- gle of land, or any other quantity of area and the length of the lines may rep- resent feet, miles, or any other units of measure. The dotted lines A, D; D, C; C, A, represent the co-ordinate triangle A, D, C, and the arc N, C, E represents the one-fourth of a circle or 90 degrees from meridian line A, D, N, to east line A, B, E, and C, A will be bearing south' 53 degrees, west, 50. The five divisions of equal distance in triangle A, C, B, are given to show the similarity of right angled triangles. Multiply the three sides of the first triangle, 10, 8, 6, by 2, and the triangle is increased to 20, 16, 12. Multiply tri- angle 10, 8, 6, by 3, and the triangle sides will be 0, 24, 18. The multiplication of the length of the three sides of any right angled triangle does not alter the d agrees of the angles; it only increases the area in proportion to the increase of the length of the three sides, and to divide the length of all the sides of a right angled triangle by the same number, decreases area without changing the degrees of the angles. Hence, to divide the length of any one of the three sides of a right angled triangle by any number that will divide without a remainder; then find the length of the other two sides of the reduced triangle by the division. Then multiply the other two sides of the reduced triangle by the same number that the first side was divided by, and it will increase the right angled triangle to its original dimension, and the number used as a d visor and multiplier will be a base. (Logrithms, have a base of 10, 100, 1,000, thtt is generally used as a basis of logrithmic sines, tangents, etc., wl i3h is a decimal basis of the radius of a circle. ) Take C as beginning of diagram No. 3; thsn lins C, A, would read south 53 degrees, west, 50. By application of the protractor to the lines, wculd show that angle A, B, C is 37 degrees, which would also be found by subtracting 53- degrees from 90 degrees. Thirty-seven degrees are the complement of 53 de- grees. A, B. would be departure east 40, and B, C, would be latitude north 30 degr- The area of triangle A, B, C, is fonnd by multiplying line A, B, by one- half of line B, C, or multiply one short side of the triangle by one-half of the other short side. See rule. The elastic scale .will ni3asure ths length cf arc N, C, rrd r.rc C, E. f.'ee rule for finding length of arc by protractor and by calculation.- DOLMAN'S PROTRACTOR. DIAGRAM NO. 4. Diagram No. 4 differs from diagram No. 3 in two particulars, viz: First, all the lines of diagram No. 3 remain inside of their circle. Second, the longest line of the triangle (bearing line) and degrees of an- gle ars given to find the latitude and departure of two short lines of the right angled triangle. In diagram No. 4 the longest lines pass outside of the circle, and are called tangent, co-tangent, secant, and co-secant, and the shortest sides never pass outside of the circle, and are given with length and the degrees of angle to find the length of the long lines. The two short sides are called sine and co- sine, and the length of tangent and secant increase of the number of degrees of sine of arc increase, and decrease in tho same way, when co-sine and tan- At an angle of 45 degrees, sine and co-sine are of equal length; tangent an-1 C3-tang3.it are. of ecju-il l?:v;lli, an:l see.vnt anil co-seoant are of equal length.' At 93 degrees t-ie bpunch of the triangle are reached by either sine or co- sin 3/a li is called infinity'. See definition of infinity. 1(5 DOLMAN'S PROTRACTOR. The radius of the circle in diagram No. 4 is taken as unity, and sine and tangent form sides of similar right angles. Latitude in diagram No. 3 corres- ponds with sine in diagram No. 4, and departure in No. 3 corresponds with co-sine in No. 4. The length of all lines of diagram No. 4 can be constructed and measured with the protractor by similar instructions as given for constructing diagram No. 3. Practice drawing tangents and secants to the radius of a circle to every five degrees of the 90 degrees of the circle, and note the rapid increase in the length of the tangent and secant when the secant and tangent angle approaches 90 degrees. B DIAGRAM NO. 5. Diagram No. 5 is given to show how to obtain distance to inaccessible objects by the right angled triangle as given in co-sines in diagram No. 4 C is first point of observation; A is first inaccessible object; D is second point of observation to object A. C is first point of observation to B, or second inac- cessible object, and E is second point of observation to object B. Required the distance from C to A and the distance from C to B and the course and dis- tance from A to B. We have a compass to give angles,* and chain to give distance, C, D, the compass says, course C, A, is south, and the line D, A is south, 18 degrees* *Mi'nsiin-l lim-imi-t always be lakcn at ri^ht ungta to the line from observation to oi,j cf. DOLMAN'S PROTRACTOR. 17 west, and the chain gives measure eight to line C, D. Now .we have a co-sine of 18 degrees and eight measurement of sine. We now place centre of the protractor on A; place one arm 'on OM line; place the other arm on 18 de- grees from OM line, move bar DD parallel until eight spaces on bar DD fills the space between the two arms. The arm on OM line from needle or center to bar DD will be 30, the dis- tance from C to A in the same unit of measure that 0, D was measured with on the ground. We find by compass that the course from C to B is south, 20 degrees west, and course from E to B is south, seven degrees west, giving a co-sine of sevejp degrees and measured line C, B is 10. Place centre of pro- tractor on B and place one arm on OM line of the protractor and the other arm seven degrees from OM line; move bar DD until 10 graduations on bar DD fill the space between the two arms. The arm on OM line from bar DD to centre will be 32, the distance from C to B; the other arin will give distance from E to B. To obtain course and distance from A to B, construct right angled trian- gles C, Ji, A, and C, E, B; place side LL of protractor on line C, A; move pro- tractor until bottom line .. DD meets A; then mark Y. Change sides of the protractor and mark Y on the other side of 'line C, A, and Y, Y is parallel to C, D. Place centre of protractor on A; move bottom line DD of protractor on Y, Y; place arm on B, and course from A to B will be south 73 degrees, wast; and, distance from A to B will be twelve on the arm. Line D, C must bs taken at right, angle to C, A, and line C, E must be constructed at right a'igle to C, B. Thus, we see that having, the length of one 'line, and two an- gles of any right angled triangle, or two lines and one angle given, the decimal scale measure protractor can give length of th# other two sides 'an'd angle, or two angles and one side. - - "^ Tns dotted lines in diagram No. 5, are given to show co-ordinate angles and opposite bearings of the diagram. To prove the angles A, C, D, and B, C, E, take same amount of distance and area of the circle, or opposite ; dir"e'ctions, that triangles C, E, a, and C, E, b take from the circle. Any course may be taken from toint : of observation. Measured line must always be constructed at right angle .to) ' object line; The names' that line takes in the different diagrams should be remembered, to prevent error in calcula- tion. Note in plotting field notes . of land opposite parallels rbust be made at every angle that is less QGT greater than 90 degrees, viz: Fir^t, to have a par- allel mark to adjust the protractor at the next angle. Second, to find latitude and departure to the angle. No survey is correct unless the lines close by latitude 'arid departure, ex- tend as far north as south, and as far east as west, called in surveying, north- ing and southing, and easting and westing. This rule should be well understood. The bearing of a right angled trian- gle given, required the latitude and departure. FIRST RULE BY PROTRACTOR. Place one arm on OM line and placa the other arm on the number of de- IS DOLMAN'S PROTRACTOR. grees of departure of the angle; move bar DD parallel to the number on the arm that is not on the OM line, and bar DD will be departure, and the arm on OM line will be latitude. The latitude and degrees of departure of a right angled triangle given, required the bearing and departure of the right angled triangle. SECOND RULE BY PROTRACTOR. Form right angled triangle on protractor, and the arm on OM line will be latitude, bar DD will be departure, and the other arm will be bearing. The departure of a right angled triangle and degrees of departure of the angle given, required the latitude and bearing of the right angled triangle. THIRD RULE BY PROTRACTOR. Form triangle as before. The first rule applies to line A, C, diagram No. 3. Rule second applies to perpendicular in triangle, diagram No. 2, and line A, B, in diagram No. 3, and tangent A r I in diagram No. 4 t and lines C t A, and C, B, diagram No. 5. Third rule applies to co-sine in diagram No. 4, etc. Line C, D and C, E in diagram No. 5 coincides with co-sines in diagram No. 4 and line C, B, in diagram No. 3. . Dolman's New Decimal Scale Measure Protractor, patent June 10th, 1890, produces length and position of all lines and degrees of angles required by arithmetic, algebra and calculus by practical object lessons scale measure, thereby solving millions of problems, by physical measurement in constructive geometry without the assistance of algebra and intricate calculus. This protractor conveys the idea of numbers and magnitude as applied to* practical architecture, mechanics, land surveying, civil engineering, naviga- tion, mine surveying, irrigation, hydrography and astronomy. TLis protractor is a complete drafting outfit for tl e student, and when made of metal and graduated to 100 to 1 inch with vernier to read minutes, it is the best and most convenient practical protractor in use. The Decimal Scale Measure Protractor gives double parallel Lines and when connected form right angles. All angles of every polygon that are less or greater than right angles must have a right angled triangle constructed or conceived to that angle before the area of that polygon can be ascertained, and the area of the constructed right angled triangle must be ascertained separate from the area of the other parts of the polygon, and added to complete the area of the polygon. When constructing polygons, with the protractor, every angle of the polygon that is lets or greater than a right angle must have a latitude and de- parture line ascertained by leaving the bearing arm on the line of the polygon., then place the other arm on the OM line of the protractor; move bar DD par- allel until the end of the line is met by bar DD. The distance between the two arms on DD will be departure, and the distance from the centre will be the latitude and departure of every right angled triangle. The latitude and departure of every right angled triangle are the two short sides of the tri- angle. DOLMAN'S PROTRACTOR. 19 When the protractor's centre is moved to the end of the line to construct another line and angle to the polygon, the bottom line DD must be placed on the departure line to preserve the parallels to the meridian or base line o: the polygon, and when the course reverses, the top of the protractor must be turned one-half around, so that the parallels may not be lost hi returning to the beginning point of the polygon. No polygon is completed until the last line meets the beginning point (called closing the survey or polygon.) The latitude and departure lines of a polygon should be indicated by dot- ted lines, and the length of the latitude and departure lines should be noted, that the area of the triangle may be computed. The latitude and departure should be on the outside of the polygon to continue the parallels with the protractor. Co-ordinate latitude and de- parture lines may be constructed on the inside of the polygon to prevent con- fusion and error by adding area to the polygon whose angles of the polygon are less than right angles. The length of either side of a right angled triangle of any conceivable length may be reduced by dividing the length of the side of the triangle by any number that will divide it without a remainder, to a number less than the graduations on the protractor; then find the other two sides of the triangle on the protractor, and multiply the two sides thus found on the protractor by the same number that the side of the large triangle was divided by. This will give the length of the other two sides of the large triangle, which is all there is in similar right angled triangles of latitude and departure, logrithmic sines, tangents, etc. Dolman's New Decimal Scale Measure Protractor produces length and position of all lines and degrees of angles required by arithmetic, algebra, and calculus by physical lines and angles, solving and proving millions of problems. The question is asked "How does the Decimal Scale Measure Protractor solve and prove an infinite number of problems?" We answer that the right angled triangle is a unit of comparison of measure between regular and irreg- ular polygons. All polygons are divisible into some number of right angled triangles of equal or different dimensions, and to multiply one short side by one- half the other short side of any right angled triangle gives the area of the triangle. The Decimal Scale Measure Protractor can be made to give the length of all three sides and the three angles to every right angled triangle if the length of one side and one angle are given.* All areas are determined either directly or indirectly by multiplying one short side of a right angled triangle by one-half of the other short side. See rule. Multiplying the length of a line by its length gives the area of a square whose side equals the length of the line, and whose area is equal to the two *The right Jingle is always understood without giving it when a right angled triangle is given. 20 DOLMAN'S PROTRACTOR. right angled triangles of that square, and the same rule applies to equian- gular parallelograms. The Decimal Scale Measure Protractor givei* double parallel lines to a, meridian or base line, and donble parallel lines forrn right angles. All angles of every polygon that are less 'or greater than ri ?ht angles must have a right angled triangle constructed to 1 that angle before the area of that polygon can be ascertained, and the area 'of the constructed right angled triangle must be ascertained separate from the area of the other parts of the polygon and added to complete the area of the polygon. When constructing a polygon with the protractor, every angle of the polygon that is less or greater than a right ailgle'must have a latitude and de- parture line ascertained by .-leaving the beasring arm on the line; then place, the other arm on the OM line and move bar DD parallel until the end of the line of the polygon is reached. Then the distance between the two arms will be the departure and the distance from the centre tb bar DD will be the lati- tude of the right angled triangle.' ' The latitude and departure of every right angled triangle is the two short sides of the triangle. When the centre of the protractor is- moved to the end of the line^to'coristrtict another side and angle to the polygon, the bottom line DD must be placed 6n the departure line last made to preserve the parallels to ihe 'meridian or base lins, and when the course reverses, the top of the protractor must be turned one -half around, so that the parallels may not bo lost in returning- to the begirining point of the polygon or survey. No survey, or polygon is correct unless the lines close by latitude and departure. The latitude and departure lines of a polygon should be designated by dotted lines, and the length of -latitude and departure should be noted, that the area of the triangle may be computed. The versed sine, or perpendicular line betweon arc and chord, is change- able in length, viz: First, the length of the versed sine is 'always equal to the difference , jn tfre length of the two longest sides of the rig-ht angled tri- angle. Second* when the length of the sine is : added to the length of lati- tude, their sum would equal the radius of their circle. Third, when the versed sine is added to the one-half diagonal of an: inscribed square, that line will equal the radius : of a circle that will describe the inscribed square, and the area of the last : circle will be double the area of the first circle; and the area of inscribed and desqribed circles can be doufcted, and the- area of inscribed and described squares doubled ad infinitum.- See diagram No. 1. ' The length of either side of a ripfht angled triangle ; bf any conceivable length may be reduced by dividing toe leiigiti of the '-'sitfe-of the triangle by any number that will clivide it with out- a. remainder to a number less than the number of graduations on the protractor; then find the other two sides cf the triangle on the protractor and multiply the two sides thus found' on the pro- tractor by the same number that the- side of the^ larg triangle was divided by' and it will give the length of the other two sides of the, large triangle, which is all there is in similar right angled triangles of latitude and departure, log- rithmic sines, tangents, etc. Dolman's New Decimal Scale Measure Protractor produces length and position of all lines and degrees of angles required by arithmetic, algebra and DOLMAN'S PROTRACTOR. 21 calculus by physical lines and angles; solving arid proving millions of problems. TEXAS LAND. MEASURE TABLE:] TABLE OF THE GEOGRAPHICAL MILES IN A DEGREE OF LONGITUDE AT EVERY DEGREE OF LATITUDE ON THE TERRESTRIAL SPHEROID, THE ELLIPTIC - ITY BEING ASSUMED 1-300. The Standard of Texas Land Meas- ure is the 10 vara chain containing 50 links. 6 2 tj inches equals 1 link. 1 vara equals 5 Ijnkg. 33> a inches equal 1 vara. 237 2-10 varas equal U B ..mile. 475 2-10 -< " , >4 " 950 4-10 " '" :",.^a " 1900 8-10 75 13-100 1000 4080 2-10 3555 5-10 "2886 2500 5000 1344 NUMBER OF f'3 K ,"fV' r 3 ' 1 4 " :CT &* 1 labor, i section. t 1. \* t side of 1 acre. i M u 1 labor. i tt " 2 3 league. i tt " U u .- t it it '** i ' t tt n u i tt " '..section DTJ/ 2kr ,*-, OTJ, . i\ J_J- ^ VARAS IN ^. c> 25,000,000 sq. varas equal 1 league. 16,666,666 2 3 sq. " 12,500,000 " " 18,333^33 <4 Ht 6,250,000 .-l,000i00 " 3,613,040 " 1,806,620 22,-iOO sq. va.equail ^section. 903726016-100 " " u ^ 4 u ' 451,630 8-100 " " " ^t * 602,173 44-100 " " " 1-6 " 5,645 " u ' 1 acre. - 4428 697-1000 acres ' A< 1 league. 177 " 1 labor. 1111 ^ sq. in. equal 1 square vara. 7 tq. It. &1C'3} 3 tq. in. 'equal Isq. vara. To reduce equatorial miles to statute miles: ItCLE. Multiply the equatorial miles by 69 1-8 and divide the produst by 60. To reduce.var^ to acres, & . RUL^.--Multiply the numbes of varas by 177 J 8 , cut of? 6 decimals from the product, the remaining figures of the product will be acres and the decimals Will be fractions of an acre ; or divide number of square varas by 5,645. Measure all lines of the diagram with the protractor to learn its use. Every School Teacher, Mechanic and Scholar should have one of thase protractors with which to practice drafting. The Wise County Protractor Publishing Co., want an agent in every city, town and county in the U. S. to sell Dolman's Protractor. Reserved territory and a liberal commission given to agents. Write for special terms to J. H. Dolman, Abilene, Texas., General Agent. The price of the protractor is SI, postage free. Address all orders for protractors to Roy B. Bradley, Abilene, Texas. Lai lLorig.|Lat.|lLong. Lat. lLong. o Mile* iSJo Miles. o Miles. .60*000 30 52004 '40 30.074 1 '59.991' 9fl 51.475 61 29 162 . 2 59.963 32 50929 28241 3 59.918 33 50 369 C3 27311 4 59 855 34 49793 64 26373 5 - r* 59 773 35 , 49 202 25 426 6 59.673 36 48.596 <66 24 472 7 59 5j,6 37 47 975 67 ! 2351 8 59421 ^38" .47-339 68 V2541 9 59 287 39 '46689 69 21565 10. ,'9095 I 6 46025 70 ; 20 581 11 58.905* 220 78 12514 19 56.751 49 39 437 79 1! 485 i 20' 56403 50 ' 38 641 80 10 452 : 21 56 038 51 37.8*4 81' 9416 5:2 55.6/)6 52 37014 82 8 378 23 5.i.258 53 36184 83 -7337 24 54.842 54 35342 84 j6*293 25 54.411 55^ 84491 5247 X6 3962 fttf 33628 86 4.199 27' 3497 57 3.' 754 ,87 8.149 28 53015 8 31 870 88 - 2 UK) 29 52 518 59. 30 1/77 89 -l.(iBO 30 52 U04 60 ; 30 074 9U O.COO *iY M . HATH A WAY'S IMPROVED TRAVERSE TABLE, WITH RULES FOR OBTAINING NATURAL SINES, TANGENTS, ANGLES, ETC Copyright, 1896, by C. F. HATHAWAY. I olman's New Decimal Scale Measure Protractor is graduated to degrees and whole units of lin- ear measure. The following rules and table are given with reference to the application of the Scale Measure Protractor to diagrams Nos. 3 a>d 4 in instructions for using the Protractor in constructive geometry and trigonometry. The table shows- latitude and departure to lour deci- mal places for linear bearing 1.00 and for angular bearings from to 90 degrees. If the angular-bearing isle s than 45 degrees the angle will be found in the 1st., 5th:. rr 9th. column Of the table ai d the lii e r bt arinp t the top r lotum column and decimal 2 K56 in the departure col - umn. Multiply .9790 by bearing 31 = 30.3490; and 2036 by 31 =6. 3116 required latitude and depar- ture. EXAMPLE SECOND Example 2nd: Required the latitude and linear bearing to g;ven linear departure 26 and the angular bearing 19 degrees and 15 minutes. Rule 2nd: In the table oppos.te to 19 degree* and 15 minutes we find decimal .9441 in the latitude column and .3297 in the departure column Divide the i iven departure 25 by .3297 =75.8265 the required bearing. Multiply 7R 8265 by .9441 = 71 5877 latitude required. EXAMPLE THIRD Example 3rd: Latitude 16 34 and>angular bearing 37 degrees and 30 minutes given: required linear bearing and departure. Rule 3r 1: iu the tubl-i op.>o-site to 17 d >g e -s a id 30 minutes in latitude column we find decimal .7984 and .6088 in departure column. Divide the given latitude 16.34 by .7934 = 20 5949 the required bearing, and multiply the liHear bearin^O 5949 1)^.6088 = 12. 5381 the required departure. t) -40U-1 W ^ />/> EXVMPLE FOURTH. K > Example 4tn : Given linear bearing 600 and linear departure 100: Required the angular bearing. Rule 4th: divide the given departure IOJ by given 4MMMM 600 =.1666. In the table we find the nearest number to the quotient to be the decimal .1650 opposite to degrees and 80 minutes, and .1693 opposite to 9 degrees and 4$ minutes, subtracting the quantities We have 15 minutes equals 0043; divide .0013 by IS minutes equals .00028666 the tabular dffe re nee for 1 miu- ute. Subtract 1650 from 1666 equals 0016 divided by .00028666 equals 5.581* minutes. .5815 multiplied by 60 equals 34.89 seconds. Combining the quantities we have 9 degrees, 3/. minutes and 34.89 seconds for the required angular bearing. EXAMPLE . FIFIH. Example 5th: First operation: Required the verse sine, (the verse sine i the difference be- tween the latitude am linear bearing) for linear bearing 31 and angular bearing 11 degrees and 45 minutes. By ruie I we find the latitude to be 30. 3490 subtract, and the difference .6510 is the ve se sine: Second operation: required co-tangent to the same linear and angular bearing The given bearing 31 becomes latitude and is worked by rule 3rd. The co-tangent is 6 4469 and theco-se<*ntxi.6650; third operation: required the natural tangent and secant to the same linear and angular tearing. Substitute the linear bearing 31 for latitude to natural tangent and secant. The angular bearing of the tangent and secant is found by subtracting' the. given angle 11 degrees and 45 minutes from 90 degrees equals 78 degree* and 15 minutes, therefore read the columns from the bottom and proceed as in rule 3rd. tangent is 149.0618, and the secant 152.25931 The latitude and departure given to find linear bearing: rule square the latitude and departure add them and extract the square root. Course' Dist. 1. Course.! 1 P * 8 * -o-HILat. ;. 1. Course.j Dist. 1. Lat. | Dep. Dep. Lat. Dep.j O 1 1 15 1.00000.0044 45 15 ; {079648 0.2630 45 id 0.86380.5038; 45 30 0000 0087 30 30 1 9636 2672 30 30 8616 5075 30 45 0.9999 0131 15 45 ! 9625 2714 15 45 i 8594 5113 15 1 9998 0175 ! 89 16 9613 2756 74 31 o! 8572 5150 59 15 9998 0218 45 15 9600 2798 45 id 8549 5188 45 30 9997; 0262 30 30! 9588 2840 30 30 8526 5225 30 45 9995 0305 15 45 9576 2882 15 45 8504 5262 15 2 9994 0349 88 17 9563 2924 73 32 8480 5299 58 15 9992 0393 45 15 9550 2965 45 15 8457 5336 45 30 9990 0436 30 30 9537 3007 30 30j 8434 5373 30 45 0.99880.0480 15 45 0.95240.3049 15 45! 0.8410 0.5410 15 3 9986 0523 j 87 18 9511 3090t i 72 33 0' 8387' 5446 57 15 9984 0567 45 15 9497 3132 46 15M 8363 5483 45 30 9981; 0610 30 30 9483 3173 30 30! 1 8339 5519 30 45 9979 0654 15 45 9469 3214 15 45 8315 5556 15 4 9976 0698 86 19 9456 3256 71 34 0! 8290 5592 56 15 9973 0741 45 15 9441 3297 45 15' 8266 5628 45 30 9969 0785 30 30 9426 3338 30 30 8241 5664 30 45 9966 0828' 15 45! 9412 3379 15 45!! 8216 5700 15 5 9962 087211 85 20 9397 3420 70 o 35 ti 8192 5736 55 15 0.99580.0915 45 15 0.9382 0.3461 45 15j 0.8166 0-5771 45 30 9954 0958 30 30 9367 3502 30 30 8141 5807 30 45 9950 1002J i 15 46 9351 3543 15 46; i 8116 5842 15 6 9945 1045 I 84 21 9336 3584 ! 69 o 36 o'l 8090 5878 54 15 9941 1089 45 15 9320 3624 45 15 S064 5913 45 30 9936 1132 30 30 9304 3665 30 30 8039 5948 30 45 9931 1175 15 45 9288 3706 15 45 8013 5933 15 7 9925 1219 83 22 9272 3746 68 o 37 7986 6018 ' 53 15 9920 1262 45 15 9255 3786 45 15 7960 6053 45 30 9914 1305 30 30 9239i 3827 30 30 7934 6038 30 45 0.99090.1349! 15 45 0.9222 0.3867 15 45 7907 0.6122 15 8 9903 1392 i 82 23 9205 3907 67 38 7880 6157 52 15 9897 1435- 45 16 9188 3947 45 15 7853 6191 45 30 9890 1478 30 30 9171 3987 30 30 7826 6225 30 45 9884 1521 15 45 9153 4027 15 45 7799 6259 15 9 9877 1564 i 81 24 9135 4067 66 39 7771 6293 51 15 9870 1607 45 15 9118 4107 45 15 7744 6327 45 30 9863 1660 30 30 9100 4147 30 30 7716 6361 30 45 9856 1693 15 45 9081 4187i 15 45 7688 6394 15 10 9848 1736 80 25 0! 9063 4226 i 65 40 7660 6423 50 15 0.98400.1779 45 15 0.90450.4266 45 15 0.76320.6451 45 30 9833 1822 30 30 9026 4305 30 30 7604 6494 30 46 9825 1865 15 45 9007 4344 15 45 7576 6528 15 11 9816 1908 i 79 26 8988 4384 i 64 41 7547 6581 49 15 9808 1951 46 15 8969 4423; 45 15 7518 6593 45 30 9799 1994 i 30 30 8949 4462 30 30 7490 6626 30 45 9790 2036 15 46 8930 4501 15 45 7461 6659 15 12 9781 2079 78 27 8910 4540 1 63 42 7431 6891 43 15 9772 2122]! 45 15 8890 4579 45 15 7402 6724 45 30 9763 2164 30 30 8870 4617; 30 30 7373 6756 30 45 0.97530.22071! 15 45 0.8850 0.4656 15 45 0.7343 0.6788 15! 13 9744 2250|! 77 28 8829 4695 ! 62 43 7314 6820 47 15 9734: 2292 46 15 8809 473S 45 15 7284 6852' 45 30 9724 2334! ! 30 30 8788 4772. 30 30 7254 6884 30 45 9713 2377] ! 15 46 8767 4810 15 45 7224 6915 15 14 9703, 2419 1 76 29 8746 4848 61 44 7193 6947 46 15 9692 2462! 45 15 8725 4886 45 15 7163 6978 45 30 9681 2504 30 30 8704 4924 30 30 7133 7009 30 45 9670 25461 15 45 8682 4962 15 45 7102 7040 15 i 15 9659 2688 i 75 30 8660 6000 60 450 7071 7071 45 0; Dep. | Lat. pL_L Dep. | Lat. 1 Dep. Lat. Dist. 1. jCourge. Dist. 1. Course. 1 Dist. 1. i ..Course." HATHA WAY'S IMPROVED TRAVERSE TABLE, WITH RULES FOR OBTAINING NATURAL SINES, TANGENTS, ANGLES, ETC Copyright, 1296, by C. F. HATHAWAY, I olmau's Now Decimal Scale Measure Protractor is graduated to degrees and whole units of lin- ear measure. The following rules an datable are given ^:it-h reference to the application of the Scale Measure Protractor to diagrams No's. 3 and 4 in instructions, for using the Protractor in constructive geometry and trigonometry. The table shows latitude and departure to lour deei- . mal places for linear bearing 1.00 and for'angular bearings from 6 to 90 degrees. Ii T. Example 1st: Given t'he Angular beari.-^gll degrees and 45 minutes and linear bearing 31; requirf d the latitude and departure of th#:triangU- Rule 1st. ; In the table opposite 11 degrees 45 minutes we find the decimal .9790 in tlyj latitud ; -column and decimal 2036 in the departure col- umn. Multiply ,9790 by bearing 31 = 30:3490; aud 2036 by 31 = 6.S116 required latitude and depar- ture. '-..' " - i - EXAMPLE ".SECCpS'p- Example 2nd: Required the latitude and linear, bearing to given linear departure 25 and the angular bearing 19 de'gree* and 15 minutes. Rule 2nd: In th^ table opposite to 19 degrees and 15 minutes we find -decimal .9441 in the latitude column and' ! .,3297 in the departure column Divide the i iven departure 25 by .3297 =75.8265 the required bearing. Multiply 75.8266 by .9441 .=? 71 5877 latitude required. EXAMPLE THIRD Example 3rd: Latitude 16 3 4 and angu.laV bearing 37. degrees and 30 minutes given; required linear bearing and departure. ' Rttle3r-l: in tli; bible opposite to 37 d ;g :e ; -s, a .'id 30 minutes in .latitude column we fiiil decimal .7934 and .6088 in departure column. Divide the given latitude 16.34 by .7934 = 20.5919 the^required bearing, and multiply the linear bearing 20.5^49 by .6088 = : 12 .53S1 the required <1< parture. ,, & J^A/^tWA (&UV EX vMPLE FOURTH. JJ "f * " Example 4th : Given linear bearing t>0j .- ojid linear -departure 100: required the angular bearing. Rule 4th: divide the given departure IOQ by given JMMlftf 600 =.1666. lathe table we find the nearest number to the quotient to be the decimal .1650 opposite to 9 degrees and 30 minutes, and .1693 opposite to 9 degrees and 45 minutes,' subtracting the: quantities we have 15 minutes equals .0043; divide .0048 by 15 minutes equals .00028666 the tabular difference for 1 min- ute. Subtract .1650 from 1666 .equals 0016 divided by .00028666 equals 5,5815 minutes. .5815 multiplied by 60 equals 34.89 seconds. Combining 'the quantities we have 9 degrees, 35 minutes and 34.89 seconds for the required angular bearing. . - EXAMPLE FIFTH. Example 5th: First operation: Required the verse sine, (the verse sine is the difference be- tween the latitude am linear bearing) for linear bearing 31 and angular bearing 11 degrees and 45 minutes. By rule 1 we find the. latitude,. to be.30.3490. subtract, nnd the difference .6510 is the ve-^esine: Second operation: required co-tangent to the same linear and angular bearing The given bearing 31 becomes latitude .and is Avorked by rule 3rd. The co-tangent is 6.4469 and the co-secant Hl.6650; third operation : required the natural tangent and secant to the same linear and angular bearing. Substitute the linear bearing 31 for latitude to. natural tangent and secant. The angular bearing of the tangent and secant is found by subtracting the given angle 11 degrees and 45 minutes from 90 degrees equals 78 degrees and 15 minutes, therefore read the columns from the bottom and proceed as in rule 3rd. tangent is 149.0618, ;aud the secant 152.2593. The latitude and departure given to find linear bearing: rule square the?latitude and departure add them and extract the square root. TTHI71 Photomount Pamphlet Binder Gaylord Bros., Ir Makers Stockton, Call PM. JAN- 21. 1 Dolman, Instruqtior s for us ing Dolman s n<.7- . -prQ-hr/v NOV b lij 1936 63970 QC100 .5 J)6 THE UNIVERSITY OF CALIFORNIA LIBRARY