VP 3 vOO. c^ -V, •^ «>' /. -> x>\^^ .0^ ^.<^^- .^^ -^^^ v' x^^' ^/ \ ^. ..\^' -^^ '■ . -'^ ■^^ c ° '^ '•^ • 0- ^^"^ J^ .J> \^ x^'^> '"■>>" .,^'^f^. '- -^^ o. .0- .^^ i^t^ vXSS3j> >" .\^ .0 5. a\ , N <- ^ -/' - ^ ^-v %-4 •is' ^S^"' (^^' -^o. ■ O' ^OQ^ o>' -«^" , _ --Sir PRODROMUS MATHEMATICAL AETS CONTAINING DIRECTIONS FOR SURVEYING AND ENGINEERING. BY AMOS EATON, A. B. & A. M., 'oniur Professor iii Kensselaer Institute, and Prof. Civil Engineering. Ten years an acting 1/and Agent, Surveyor, and Engineer ; while pursuing the profession of Law. Member of the Amer. Geol. Soc— ofPhil. Acad. Nat. Sci— of N. York Lye. Nat. Hisi.,&c. TROY, N. Y. PUBLISHED BY ELIAS GATES. ALBANY, O. STEELE ; NEjW-YORKJ' ROBINSON, PRATT & CO. ; PHILADELPHIA, HENRY PERKINS. TOTTLE, BELCHER AND BURTON, PRINTERS. 1838. If Entered according to Act of Congress, in tlie year J838, by iJio pioprietor, Elias Gates, in the Clerk's Office of tiie District Court of the Nortliern District of New- York. 3 3 ^ t- ^-J?74.6. F R E F A C K . This book is chiefly made up of selections from a mass of hete- rogeneous materials, which I have been depositing in my common journal for more than thirty years — some of which, howevei', 1 published in 1830, under the title " Art without Science." I may add, that I had published a very small treatise under the same title, in the year 1800. Students have made use of manuscript copies from my notes, for a kind of guide in their course of exercises, for the last four years. Examiners, appointed by the Patron of this Institute, have followed them, mostly, for the same period. Though it is offered as the Prodromus* of a full treatise on Ma- thematical Arts ; I have progressed too far on the way to the iourne of three score and ten, to give any assurances. I have materials enough to complete the object ; but " there is a point by nature fixed, whence life must downward tend." Logarithms are not used in this book for purposes of calculation. It is a mistake to suppose that logarithms expedite calculations in trigonometry, in common applications of it. The tedious processes of multiplication and division, when we use natural sines, are over- balanced by the trouble of looking out logarithms and accommo- dating them to the various cases. This opinion is to be proved or disproved by trial alone. But in a long tedious process of several days labor, logarithms are generally useful. Algebraic expressions are not used ; because they are unneces- saiy, and very few are sufficiently versed in algebra to apply them to advantage. In truth, our fives are too short to devote much time to speculative mathematics. In past ages, when the science of nature was in its infancy, more time could be devoted to " mere tricks to stretch the human brain," than in this day of astonishing developements of nature's wonders. * Prodromos, Greek, fore-runner. Every teacher of experience knows, that the only successful mode of instruction is that which interests the student. Also that it is^ exceedingly ditTicult to excite interest by a blindfold course, whose object is not perceived. In learning land surveying, the student should always survey, under the mechanical direction of a teacher, before he studies the science of surveying. He should take latitude and longitude, and learn the use of the necessary instruments, before he devotes a day to the theory of lunar observations, &c. He will then perceive the object of his closet studies, and hear, understand- ingly and with dehght, the lectures of his teacher. Even the common proportions of a triangle should not be introduced to a student's mind, until the teacher has directed him in the measurement of the lines and angles in true-earnest application. It is scarcely more absurd to attempt to theorize a blacksmith's boy into horse-shoeing, than to attempt to make a practical mathematician without out-of- doors practice. But the extreme of absurdity is most emphatically exhibited by putting books into students' hands, written in a language which they cannot understand. If they must read a language which is new to them, they must have time to learn it. An honest teacher ought, in such cases, to make the students or their guardians, understand that they are not to study mathematics, until they have devoted a year to the study of a new language, called algebra. In this small work, an attempt is made to enable a common-sense farmer, mechanic, merchant, or other man of business, who is but an ordinary arithmetician, to become sufficiently qualified for the business concerns of life, as a practical mathematician. But he must be shewn the use of mstruments ; as it is an idle waste of time to attempt to learn their use from books. AMOS EATON. Rensselaek Institute, Troy, March, 1838. POCKET SCALE Of Natural Sines, Chord Line, and Equal PaHs. A six inch pocket ruler has always been considered as essential to a mechanic. To a mathematician, or even an ordinary traveller, dzc., such a measure should furnish the necessary scales of what we need most. A scale of equal parts of an inch in tenths, and a diagonal inch for hundredths, are the most important. A line of chords, of about a three inch sweep of sixty, is generally deemed next in importance. Geometrical trigonometry may be wrought by these two scales. But our situation is not always (nor even at one occasion in one hundred) such, that we can sit down at a table, and use the scale and dividers accurately. But we can make calcula- tions with natural sines, while riding in a stage, or sitting at the theatre, or a concert. But this operation demands a table of natu- ral sines. A book, then, of many pages, must be carried in our pockets. To obviate this difficulty, I have prepared the annexed Abridged. Table of Natural Sines. Particular directions for its use are hereunto subjoined. A six inch ruler, 2 inches wide, will soon be made in Troy, N. Y.. containing the diagonal scale of equal parts, the chord line, and the abridged table of natural sines, carried out to sines of degrees and minutes. It will always be found at bookstores, where this Prodromus is sold, after the artist has com- pleted his instruments for making the ruler. ABRIDGED TABLE OF NATURAL SINES. At page 31, a table with this heading is printed. It was hurried into that form for cases where perfect exactness in the minutes should not be required. It was afterwards discovered, that by ex- tending the augments, degrees and minutes might always be calcu- lated and used, when books, containing full tables, were not at hand. DIRECTIONS For using the Abridged Table of Natural Sines, here inserted — also^ on the Pocket Ruler. 1. If the sine of a whole degree, or of a half degree (30 minutes) is required, it is found against such degree or half degree (that is, such degree and 30 minutes.) 2. If the sine of any number of minutes, less than 30 degrees, is required, proceed as follows : Find the number in the column of augments, against the last degree preceding the number of minutes, whose sine is required. Multiply this amount by the number of minutes, and add the product to the sine of the said preceding de- gree ; setting the right hand figure one place to the right of the sine. The sum will be the sine of the degrees and minutes required. 3. If the sine of any number of minutes more than 30, is required, proceed thus : Find the number in the column of augments, against the last 30 minutes preceding the minutes whose sine is required. Multiply this augment by the number of minutes exceeding said 30 minutes ; and add the product to the said sine of said preceding 30 minutes — setting the right hand figure one place to the right of the sine. The sum is the sine of the degrees and minutes required. 4. In an operation in trigonometry (where natural sines are used, see sec. 39) if the answer is in sines, find the degrees and minutes as follows : Find in the table the nearest degree or half degree, next less than is required. Subtract that sine from the said an- swer, and divide the remainder by the augment set against the de- gree or half degree ; which will give the additional minutes. These added to said nearest degrees and minutes, give the true degrees and minutes required. 5. In most cases of ordinary practice, the subdivisions of a de- gree into six parts (10 minutes each) will be sufficiently accurate. Every sixth division (that is, every 10 minutes) requires a very simple application of the augments. Ten and 40 minutes require the augment once merely, without extending a figure to the right of the sine — 20 and 50 minutes require the augment doubled, and not extended to the right of the sine. ABRIDGED TABLE OF NATURAL SINES. 3 = De- s. c-s. c-s. s. De- i'-3 S - De- 3. c-s. □-S. s. De- 1 3 — < 1 grees. grees. < 1 < 1 grees. grees.L- g 291^ 0.00 0.00000 1.00000 90.00 267 23.001 .390721 .92050 67.00 112 291 .30 .00373 .99996 .30 1 266 .30 .39875 .91706 .30 114 291 1.00 .01745 .99985 89.00 3 265 24.00 .40674 .91355 66.00 117 291 .30 .02618 .99966 .30 6 264 .30 .41469 .90996 .30 120 291 2.001 .03490 .99939 88.00 8 263 25.00 .42262 .90631 65.00 121 290 .30 .04362 .99905 .30 11 262 .30 .43051 .90259 .30 123 290 3.00 .05234 .99863 87.00 13 260 26.00 .43837 .89879 64.00 126 290 .30 .06105 .99813 .30 16 259 .30 .44620 .89493 .30 128 290 4.00 .06976 .99756 86.00 18 257 27.00 .45399 .89101 63.00 130 290 .30 .07346 .99692 .30 21 556 .30 .46175 .88701 .30 133 290 5.00 .03716 .99619 85.00 24 256 23.00 .46947 .83297 62.00 135 290 .30 .09585 .99540 .30 26 255 .30 .47716 .87882 .30 137 290 6.00 .10453 .99452 84.00 28 1 253 29.00 .48481 .87462 61.00 140 289 .30 .11320 .993.57 .30 31 252 .30 .49242 .87036 .30 141 28S 7.00 .12187 .99255 33.00 34 251 30.00 .50000 .86603 60.00 143 288 .30 .13053 .99144 .30 37 250 .30 .50754 .86163 .30 145 287 8.00 .13917 .99027 82.00 39 248 31.00 .51504 .85717 59.00 147 287 .30 .14781 .93902 .30 41 246 .30 .52250 .85264 .30 151 287 9.00 .15643 .98769 81.00 43 245 32.00 .52992 .84895 58.00 153 286 .30 .16505 .98629 .30 46 244 .30 .53730 .84339 .30 155 286 10.00 .17365 .98481 80.00 49 243 33.00 .54464 .83367 57.00 157 286 .30 .18224 .98325 .30 52 242 .30 .55194 .83389 .30 160 235 11.00 .19081 .98163 79.00 54 240 34.00 .55919 .32904 .56.00 162 284 .30 .19937 .97992 .30 57 238 .30 .56641 .32413 .30 163 283 12.00 .20791 .97815 78.00 59 237 35.00 .57358 .81915 55.00 165 283 .30 .21644 .97630 .30 62 236 .30 .58070 .81412 .30 167 283 13.00 .22495 .97437 77.00 64 234 36.00 .58779 .80902 54.00 170 282 .30 .23345 .97237 .30 66 232 .30 .59482 .80386 .30 172 282 14.00 .24192 .97030 76.00 69 230 37.00 .60182 .79364 53.00 174 281 .30 .25033 .96815 .30 72 228 .30 .60876 .79335 .30 176 280 1.5.00 .25332 .96593175.00 74 227 38.00 .61.566 .78801 52.00 178 280 .30 .26724 .96363 .30 77 226 .30 .62251 .78261 .30 180 279 16.00 .27564 .96126 74.00 79 225 39.00 .62932 .77715 51.00 182 276 .30 .28402 .95832 .30 81 223 .30 .63608 .77162 .30 184 277 17.001 .29237 .95630 73.00 83 222 40.00 .64279 .76604 50.00il86 277 .30 .30071 .95372 .30 86 220 .30 .64945 .76041 .30 137 276 18.00 .30902 .95106 72.00 88 218 41.00 .65606 .75471 49.00 139 275 .30 .31730 .94832 .30 91 216 .30 .66262 .74896 .30 191 274 19.00 .32557 .94552 71.00 93 215 42.001 .66913 .74314 48.00 193 274 .30 .33381 .94264 .30: 96 213 .30 .67559 .73728 .30 195 273 20.00 .34202 .93969 70.00 93 211 43.00 .68200 .73135 47.00 197 272 .30 .35021 .9.3667 .30 101 209 .30 .68835 .72537 .30 199 271 21.00 .3.5637 .93358 69.00 103 207 44.00 .69466 .71934 46.00 201 270| .30 .36650 .93042 .30 105 206 .30 .70091 .71325 .30 203 269 22.00 .37461 .92718 68.00 107 205 45.00 .70711 .70711 45.0C 205 2681 .30 .38268 .92388 .30 110 .30 .71325 .70091 .3C 0^ Students are not to apply the above Table until they have studied sections 31 to 40, inclusive. CORRECTIONS TO BE MADE WITH THE PEN. Page 13, top line — "seventeen" change to " four." Page 31 — the augments are better on page vii. Page 80, sec. 141 — "foot" change to "inch." Page 91 — change places of the words "Top" and "bottom." Page 100, 3d line— after " Station" read "No. 10." Page 100, 3d line — instead of "moved up his instrument" read "went with the Targetman." Page 100, 19th line— for " Sec. 000" read " Sec. 227." Page 100, 23d line— for " 740" read "760." Page 103, 1st line — for " Station" read "two stations." Page 111, last of sec. 216 — between "the" and "ordinate" interline "square of." Page 114, sec. 224 — " chair" change to " bench." Page 122, sec. 253, middle line — strike out " the square of." Page 133, sec. 288, near the end — "to the root add" change to "from the root subtract." REFERENCES. The student is referred to last part of the book, where wood-cut figures are described, for continuations of several sections of importance. First is from section 198 to 200, extending and illustrating by wood-cut figures, the method of cnlcvbiting rail-road curves. Second is from sections 211 to 214, extending and illustrating by wood-cuts, the method of calculating ordinates, for offsets from secondary chord lines, one hundred feet each. Third is from sections 224 to 226, extending and illustrating by wood-cuts, the method of calculating excavations and embankments. This text-book is not limited in its object to students in surveying and engi- neering. Not more than 43 pages are, exclusively, devoted to them. In it will be found those rules and directions for calculations, which are essential to bverj correct student in Geography, Astronomy, and Natural Philosophy; also to all classes of readers, who wish to understand what they read. Teachers of female institutions, and of common academies, are requested to look over the contents, and consider the manner in which subjects are treated. Such institutions may omit — 1. Practical Land Surveying, from page 47 to page 74. 2. Running out Rail-Roads, from page 95 to page 111. All this treatise, excepting the above excepted 43 pages, should be studied and illustrated with practice by every student, who is presented to the public as tolerably educated. The common practice of introducing a system of elementary rules and de- monstrations, so to cumber a small practical treatise as almost to exclude the professed object, has always appeared to be absurd. Elementary treatises, executed in a style of excellence which cannot be surpassed, are to be found 4 in almost every book-store. To them the learner is referred; and will not be taxed with the reprint, and expense of copy- right, for the sake of swelling a small work into a large one. Gibson's System of Surveying, for example, a work of almost 500 pages 8vo, contains less than 100 pages, which are devoted to the professed object of the work. It is hoped, that the learner will find nothing in this, which is useless in aid of his proposed object. With pleasure I acknowledge my obligation to State engineer Holmes Hutchinson, Esq., for his iiistructi/e explanations in answer to my numerous 1 inquiries for the last half dozen years. To engineer Wm. C. Young I am also indebted for much useful information on the construction of rail-road woiks; as exemplified and explained by him at the extensive works in Sche- nectady. For the latest and most approved method of rail-road surveying, in- cluding staking out, running curves, measuring for excavations, &c., students are referred to articles furnished by engineers Sargent and Evans. I vpill take this opportunity to say, that without exceptions, every practising engineer with whom I have had any intercourse during the existence of the Engineer department at this Institute (four years) has manifested a strong desire to aid its progress and extend its influence. CONTENTS. This table of contents is constructed for the convenience of examiners. As the by-laws of this Institute forbid all persons concerned in the instruction of students giving any opinion on the subject of their qualification, and as each board of examiners is made up of gentlemen who are wholly disconnected with the school; it was supposed, that few would be willing to devote sufficient time to each subject to extract the essential points in it. Therefore each item of the contents is made to contain what appears to be sufficient to give the student a fair clue to all that is expected from him. N. B. The word Explain, is supposed to be prefixed to every subject, in the imperative mode. In numbering sections, inclusive is understood. ARITHMETIC. No. of Secliona. 1. Three elementary operations with number. Addition, Se- paration, . . . . . 1 to 5 2. Notation, . , . . , 6 to 12 3. Common characters, .... 13, 14 4. Decimals, Addition and Subtraction, . . 16, 17 5. Multiplication and Division, . . . 18, 19 6. Bringing compound expressions to decimals, or decimals and integers, ..... 20 7. Rule of three, ..... 21 8. Roots and Powers, Square root, . . . 22 to 26 9. Cube root, . . . . . 27 to 29 10. Roots of higher powers than cube, ... 30 TRIGONOMETRY. IJ. Angles and triangles, 12. Trapezoid, and triangles between parallels, 13. Sines, line of chords, degrees ia triangles, 14. Geometrical trigonometry, 15. Proportions of sides and angles of triangles, IG. Square root and rule of three applied to trigonometry, 17. Natural Sines, .... 18. Table of Natural Sines, 31,32 33(1,2) 33 (3 to 6) 34 35 to 37 38(1,2) 39 40 to 42 No. of Sections. 19. Operations in trigonometry when two sides of a rightrangled triangle are given, .... 43 20. Also when one side and two angles are given, . . 44 21. Also when two sides and an angle opposite to one of them are given, p . . . • 45 22. Also when two sides and their contained angle are given, • 46 23. Also if one leg of a right-angled triangle, and the sum of the other leg and hypothenuse, are given, . , 47 24. Also when the three sides of a triangle are given. , 48 MENSURATION. 25. Parallelogram (or rectangle) triangle, polygon, and circle, 52 (1, 2, 3, 4) 26. Periphery of a circle, diameter, and area, . . 52 (5, 6, 7) 27. Length of an arc, sector, and segment, . . 52 (8, 9, 10) 28. Area of an oval, .... 52(11) 29. Superficies of a prism, cylinder, pyramid, and cone, . 52 (12, 13) 30. Area of a parabola, and superficies of a globe, . 52 (14, 15) 31. Solid contents of a cube and parallelepiped, . . 53 (I) 32. Solid contents of a cylinder, prism, and wedge, . 53 (N. B.) 33. Solid contents of a globe, pyramid, and cone, . . 53 (2, 3) 34. Solid contents of the frustrum of a cone and pyramid by two methods, . . . . . 53 (4, 5, 6) 35. Guaging by the double cone method, . . . 53 (7) 36. Guaging by the common formula, . , . 53 (8) 37. Tonnage of vessels, . . , , 53 (9) LAND SURVEYING. 38. Four kinds of surveying, . . . 54 to 58 39. Field surveying, . . . . . 59 to 62 40. Preparations for farm surveying, putting needle, chain, tal- lies, &c., in order, . . . . 65 to 67 41. Taking elevations and depressions by Kendall's tangent scale, ...... 71 42. Fixing starting boundary and taking a course with the com- pass, . . . . , 74, 75 43. Making offsets, ..... 78 44. Taking distances and heights of objects not on the line, , 80 45. Running a random line and making an extemporaneous cal- culation, . . . . . 81, 82 46. The method of keeping the field book, . . . 89 47. Plotting and triangular casting, . . , 91 to 95 48. Reducing a field to a single triangle, , . , 96 49. Trapezoidal method,' . . . , 100 to 104 50. Road surveying, . . . . 110, 111 No. of Sections. 51. 52. 53. 54. 55. 56. 57. 58. 59, 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. Harbor surveying by base lines on shore and intersections of lines of bearing, .... STATICS AND DYNAMICS. Distinctions between Statics and Dynamics, Hydrostatics and Hydrodynamics, Velocity of falling bodies, Weight of water compared with measure, Specific gravity of liquids as shewn by Baume's areometer, and his zero points, . . , . Taking specific gravity of solids for estimating their solidity, Method of demonstrating that water pressure depends on its height, ..... Velocity of spouting fluids increasing as the square roots of their heads above the point of effusion, Method of proving that atmospheric pressure holds water together in the liquid state, without which it would be- come vapor at 67 degrees of Fahrenheit, Method of determining how high to place the lower valve of a common pump, .... Gonatous forces applied to arches, bridges, &c., in general, MECHANICAL POWERS. Lever and its modifications, Inclined plane and its modifications, ARCHITECTURE. Pillars and parts of pillars, Orders of architecture, .... Miscellaneous structures, 122 to 124 127 to 130 131, 132 133 134 135 136 138 to 140 141 142, 143 145 to 149 151 153 154 to 156 157 to 161 164 RAIL-ROADS, &c. Three kinds of survey — Extemporaneous, Preliminary, and Definite. Extemporaneous traverse across a mountainous district with barometer and compass, preparatory to a preliminary survey, .... Hutton's formula for calculating barometrical height, Gregory's formula. Latitude and longitude surveys of rail-roads and canals of great extent. Taking latitude, . - . . Longitude by Jupiter's moons, Longitude by lunar observations, Finding the breadth of a degree of longitude at any degree of latitude, ..... 166, 167 168,169 170 of 171 to 173 174, 175 (note) 176 . 177 178 6 No. of Sections. 75. Party for a preliminary survey of a rail-road, and the com- mencement of the survey with the compass, . 179 to 183 76. Continuance of the survey with the level, and the method of keeping field notes, .... 184 77. Definite rail-road survey, . . . 186 to 188 78. Staking out, ..... 190,191 79. Pencilling curves upon a plotted traverse, and finding the radius and central angle, . . . 198, 199 60. Preparing for staking out the arc into 100 feet chords, . 200 to 202 81. Finding the starting point and setting the compass upon the first point for deflexion from the tangent, . . 203, 204 82. Preparing for, and running, on the general chord line and running ofisets (or ordinates) to the staking points on the arc, ..... 83. Running and staking from one end of the arc, 84. Making a hundred feet table of ordinates^ 85. Changing the arc of a circle to the arc of an ellipse, 86. Comparing rail-road curves, 87. Convexity of the earth causing a falling below the level, EXCAVATIONS AND EMBANKMENTS. 88. Finding the solid contents of earth to be excavated, &c., by six areas of a prismoid, .... 89. Taking cross areas by applying the trapezoid, and triangles between parallels, .... CANALS. 90. Displacement of water when boats are moved in narrow and wide canals, proving the advantage of wide canals, . 233, 234 91. Boats constructed with a view to the principle of the in- clined plane and wedge, 92. Technical terms applied to canals and locks, 93. Calculating supply of water for a canal, 94. Calculating the time of filling or emptying a lock, ROADS IN GENERAL. 95. Shady trees, resting places, zigzag roads, dugways, 96. Calculating the angle of friction of carriages on roads, 97. Location of bridges, string pieces, and placing supports, WATERWORKS. 98. Accelerating and retarding forces, 99. Modifications of the two elementary forces, and their six- fold state, .... No. of Sections. 100. Formula for pipes when velocity and quantity discharged are sought, ..... 286 101. Formula for pipes when the diameter of the pipe is sought, 287 102. Formula for open canals when velocity and quantity are required, ..... 288, 289 103. Atmospheric pressure as influencing springs, &c., in pipes varying according to the height of mountains, &c., . 291 104. Method of talung the height of the atmosphere at the point where it ceases to be dense enough to reflect light, . 292 105. Calculating the time between sunset and setting of twi- light, as used in the above calculation, . . 293 106. Aqueous vapor in fogs, clouds, and the five forms of regu- lar clouds, ..... 295, 296 107. Three forms of disconnected clouds, . . . 297 108. Taking the height of clouds, . . . . 298 WATER-POWER, APPLIED TO MILLING, &c. 109. Duty of an engineer, as to water power, confined to out of doors work, . . . . . 299, 300 110. Calculating supply of water, . . . 301 to 306 111. Calculating supply of water by a falling sheet of water, called the weir calculation, . . . 307 to 311 112. Efficiency of water power at the bottom of flumes, . 312 to 314 113. Water acting on undershot wheels, . . . 315 114. Standard of efficiency of water-power taken from Troy mills, N. Y., . . . . . 317 to 319 115. Efficiency of water-power, calculated from the square feet of the faces of mill-stones passing over each other, . 320,321 116. Poestenkill mills as standards for calculating the general power of water, .... 322 117. Calculating supply of water by the water-sheet pitch, or the weir method, ..... 323, 324 118. Example of triturating surfaces of miU-stones, . 325, 326 TOPOGRAPHY. 119. Definition, and first steps in approximating surveys, . 327, 328 120. A topographical survey in miniature, . . 329 to 331 121. Extensive topographical surveys, . . . 332 to 334 MATERIALS FOR CONSTRUCTION. 122. Natural division of the subject, . . . 335,336 123. General illustrations of inorganic and organic materials, 337, 338 124. Geological alphabet, .... 339, 340 No. of Sections. 125. Arrangement of rocks geologically, or in regular series, . 341 126. Exhibition and description of geological strata, 127. Class of primitive rocks, .... 128. Class of transition rocks, .... 129. Class of loicer secondary rocks, 130. Class of upper secondary rocks, 131. Class of tertiarj/ earths, ...» 132. Subordinate series, or the red sandstone group, 133. Anomalous deposits, .... USEFUL ROCKS AND CEMENTS. 134. Marble, freestone, flagging, &c., 135. Millstone, grindstone, whetstone, 136. Hornblende, granular quartz, argillite, . , 137. Cements of lime, gypsum, «S:c., TIMBER MATERIALS. 138. Oak timber, . . . . , 139. Strength and durability of timber, IRON MATERIALS. 140. Three kinds of iron, . . . , 141. Application of kinds of iron, j . . . MATHEMATICAL ILLUSTRATIONS. Before studying this treatise, students must have been sufficient- ly exercised under teachers' dictum, with whole numbers and decii mals, in Addition, Subtraction, Multiplication and Division, and the Rule of Three. Nothing more is required ; neither is it profitable to detain students with Compound Arithmetic, Vulgar Fractions, or Algebra, until they are made acquainted with the most useful parts of the Mathematical Arts — particularly the general applications of superfices, solids, and trigonometry, to common business concei'ns. This I assert ; and my assertion is founded on forty years' experi- ence. And this rule applies to merchants' clerks and others, whose operations are merely arithmetical. The power of numbers, once understood, applies to all cases alike. Therefore the mere land surveyor makes a better book-keeper by one month's practice, than the student in mere book-keeping does in a year. The reason is manifest. The surveyor takes, necessarily, a scientific view of the power of number; while the student in book-keeping takes a parrot-Uke rotine of artificial forms. The former is governed by sound reason--the latter is led blind-fold by authority. He is a mere machine — but the mathematician is disciplined as an intellect- ual being. Alcohol and Algebea are Arabic names. Alcohol is a power- ful agent, of vast importance. But its abuses render it a curse. Algebra is a powerful entering wedge in Mechanics, and of great importance in the concise expressions of valuable formulae. But a kind of affectation of technical learning has so far obscured the mathematical arts with algebra, as to render it an absolute nuisance. In this little treatise, algebraic formulce are translated into fair English. Biot, (a most distinguished French Philosopher,) after thirty years' experience, as teacher in the algebraic mode of ex- pression, prepared a System of Natural Philosophy, totally divested of all such technical obscurities. His authority, supported by sue cess, has, in a great measure, revolutionized the course of mathe- matical learning in France. 2 10 ELEMENTARY OPERATIONS WITH NUMBERS. Sec. 1. The science of numbers, called Arithmetic or Mathema- tics, may be resolved into three elementary operations : Addition, Separation and Notation. Addition. Sec. 2. This operation consists in uniting individuals into groups, masses, or sums; as the arranging of 100 soldiers into a group, called a company — uniting 196 pounds of flour in a mass, called a barrel — uniting the value of 100 cents into a silver coin, called a dollai' — or uniting in one sum a sufficient number of feet of plank for laying a floor in a room of given length and breadth. Sec. 3. When several additions are performed by one operation, w^e distinguish this modification of addition by the descriptive name, Multiplication: as $12 paid to each of 7 laborers must be added seven times to find out the whole sum to be paid, thus : 12+12+12 +12+12+12+12=84. But we may learn by rote to add thus : seven times twelve equals 84. This rotine metjiod of adding is called Multiplying. Separation. Sec. 4. This operation consists in taking part from the whole ; or separating smaller portions of masses, or numbers, from the larger. Individuals of groups are separated from each othei*, or distributed into smaller groups. The operation is called Subtrac- tion, or Division, according to its peculiar application. All results may be produced by Subtraction; but Division is more expeditious when a separation into numerous parcels or parts is required, and particularly when the proportional parts are to be ascertained, from given data, for proportional separation. If 1281 dollars are to be equally divided among 61 laborers, we divide the dollars by 61, by a tabular rotine, called Division, thus: 61)1281(21— giving $21 to each. 122 61 61 00 11 Sec. 5. The same result will be produced by perpetually sub- tracting 61 from 1281, and counting up the number of subtractions, thus : §1281 854 427 61 1st. 61 793 8th. 61 15th. 1220 366 61 2d. 61 9th. 61 16th. 1159 732 305 61 3d. 61 10th. 61 17th. 1098 671 244 61 4th. 61 11th. 61 18th. 1037 610 183 61 5th. 61 12th. 61 19th. 976 549 122 61 6th. 61 488 13th. 61 20th 915 61 61 7th. 61 14th. 61 21st. 854 carried up. 427 carried up. 00 As 61 can be subtracted 21 times from $1281, each of the 61 laborers will have $21, as given by dividing by 61. Notation. Sec. 6. The operation of setting down or recording numbers. Numbers are expressed by Roman letters, or by Arabic figures. But Roman letters are never used in the process of calculations. They are very convenient for expressing the numbers of large divisions which are to be subdivided : such as Classes of Plants, expressed in Roman letters, which are subdivided into Orders, and expressed in figures. Sec. 7. The Roman letters used for expressing numbers are, I for one, V for five, X for ten, L for fifty, C for a hundred, D for five hundred, M for a thousand. When these letters are joined in a horizontal row from left to right, with the smallest valued letter at 12 the right, that is added to the larger. But if the smaller valued let- ter is set on the left of the larger, it is subtracted. Thus, VI stands for six, and IV for four; XI for eleven, IX for nine; LX for sixty, XL for forty; CX for one hundred and ten, XC for ninety. No letter, however, but I, X and C, is used by us in this manner as a subtrahend. Sec. 8. The figures are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. When join, ed in a horizontal row, they begin their value with the right hand figure ; and the figure, added on its left, stands for ten times its value when alone : the next for one hundred times as much : the next for one thousand times, and so on. In the notation of figures they are pointed off in threes, from right to left. Sec. 9. The element of the first three numbers is unit — of the second three, is thousand — of the tliird three, is million — of the fourth three, is billion — of the fifth three, is trillion — of the sixth three, is quadrillion — of the seventh three, is quintillion — of the eighth three, is sextillion — of the ninth three, is sepiillion — of the tenth three, is octillion — of the eleventh three, is nonillion — of the tivelfth three, is decillion — of the thirteenth three, is undecillion; and so on, indefinitely, following the Latin numerals, with the same Anglicising termination. 01 CO 1 § 1 1 rA 1 1 1 1 n •■^ 00 2 . o a, ^ ^ CI ^ s Cnii W d, ^ v< tJL, o -S O O o .2 ° o eds f— ions. ° ^ o ° =^ .^ m 1 signifies that the number following it, is to be multipUed into something preceding, as 124x12, shews that 12 is to be multiplied into 124, making 1488. Horizontal line — , (called minus,) signifies that the number fol- lowing it is to be subtracted from something preceding, as 729 — 36 shows that 36 is to be subtracted, leaving 693. Horizontal dotted line ~^, signifies that the number following it, is to be used as a divisor for dividing some dividend that precedes it, as 806-4-26, shews that 26 is to be used as a divisor for the divi- dend 806, making 31. This dotted line is often omitted, and the figures set down like an impi'oper vulgar fraction, thus, y^. Parallel lines =, signify that the number or numbers following them, equal something preceding, as 17x5x4=:340. Inverted figure seven \/, signifies that what follows it, requires the square root to be extracted. Figure 3 over it imphes that the a cube root should be extracted, &c., as \/27 requires the cube root to be extracted, making 3. N. B. Students must be exercised in notation, until they are familiar with the right application of the characters, and arrange- ment of figures. Decimals. Sec. 15. Decimals are parts of mtegers (whole numbers,) point- ed off" to the right by a period. The first figure expresses tenths of an integer, the second hundredths, and so on, diminishing ten fold at every figure. Several figures of decimals express a general 15 proportion of an integer, as 9.6345 inches expresses nine inches and 6345 ten thousanths of an inch. But the most common as well as most convenient mode of expression is, to read off the figures separately after the point — thus : nine, point, six, three, four, five. In reading off Natural Sines, or Logarithms, this method is always to be adopted. Addition of Decimals. Sec. 16. Decimals, or integers and decimals together, are al- ways added like whole numbers. But in setting them down for adding, points must be set under each other in a column ; and let the figures on both sides of each point stand at uniform distances. This will cause some lines to project further to the right and left than others. Of course care is required in footing up the figures in their proper columns. Examples in Addition of Decimals. 76543.6201 63421.6203 360.2 9. 2.76423 243.67861 4239.621 27. 7.3 3.00001 36012.00034 789.9 117165.50567 64493.19892 Subtraction of Decimals. Sec. 17. Decimals, or integers and decimals together, are al- ways subtracted like whole numbers. But in setting them down for adding, the point in the subtrahend must be set directly under the point in the minuend ; and the figures each side of the point stand as directed in addition. Examples in Subtraction of Decimals. 63967.9342 80001. 201.7 2.0096 63766.2342 79998.9904 16 Multiplication of Decimals. Sec. 18. Decimals, or decimals and integers together, are always multiplied like whole numbers. But after multiplying, care is re- quired in placing the point between the integers and decimals. The rule is, to point off for decimals just as many places of figures, as are pointed off in both the multiplier and multiplicand ; leaving for integers all the rest, if any. Examples of Multiplication of Decimals. Diameter of a circle, 643.231 inches. Formulae for circumfer- ence, 3.1416 inches. 643.231 (operation 3.1416 omitted.) 2020.7745096 Four decimal places in the multiplier, and three in the multipli- cand, require seven to be pointed off. Therefore the circumference is 2020 inches and 77 hundredths, if hundredths come near enough. But if exactness is required, say 2020 inches, and point .7745096 decimals of an inch. Area of a square embracing a circle is 5402064.201 Formula for reducing a square to a circle, .7854 4242781.2234654 Four decimal places in the multiplier, and three in the multipli- cand, require seven to be pointed off. Therefore the area of the circle is as above — that is, 4242781 square inches, and 22 hun- dredths, if hundredths come near enough. Or, point .2234654 deci- mals of an inch. Division of Decimals. Sec 19. Decimals, or decimals and integers together, are al- ways divided hke whole numbers. But, after dividing, care is re- quired in placing the point between the integers and decimals. The rule is, to point off for decimals in the quotient so many places of figures, as to make the number pointed off in the divisor and quo. 17 tient, just equal tjie number pointed off in the dividend alone. But if there are not as many pointed off in the dividend as in the divi- sor, cyphers must be added to the dividend (beyond the point) to make the number equal. And if an equal number does not carry on the decimals of the answer far enough for the required exact- ness, more may be annexed — always applying the rule for pointing off, as before stated. N. B. We often proceed by adding decimals, in carrying on the decimals in the quotient to greater extent than those of the divi- dend will admit. In such cases, all the added decimals must be counted as if they had previously been added to the dividend. Examples in Division of Decimals. The area of a circle is 4242781.2234654 inches; divide by the formula .7854, which will give the area of a circumscribing square : .7854)4242781. 2234654( (operation 5402064.201 omitted.) •« ' Here are four places pointed off in the divisor and seven in the dividend. Now there must be pointed off in the quotient, which added to those of the divisor (being all the divisor) equal those of the dividend. The answer then is, that the circumscribing square made about the given circle, contains 5402064 square inches, and point .201 decimals of a square inch. Sec. 20. Bring compound expressions to Decimals. In all cases where measures, weights, or values of any kind, are required to be brought into decimal expressions, make a vulgar fraction express- ing the proportions. Then divide the numerator by the denomina- tor ; adding cyphers to the numerator as far as may be required. Bring £16 7s. and 4d. to the decimal of a pound: 7s. and 4d.= 88d. A penny is the 240th of a pound. Then seven and four- pence is ^Vo of a pound— 240)88.0000(.3666. Answer is £16. 3666. Note: this results in a circulating decimal ; as it will for- ever give 6, and is ever approximating the truth. Rule of Three. Sec. 21. When a value is set upon one article, it is self-evident that the value of any number of articles may be found, by multiply. 18 ing the value of one by the number whose whole value is required. As, if one square foot of ground costs 60 cents, and a house lot may be had at the same price per foot, which contains 1800 feet, it is manifest that the whole lot will cost 1800 times 60 cents, or 1080 dollars. This stated, according to the form of the Rule of Three, will stand thus : 1 : $0.60 :: 1800 : $1080.00. Here the second and third numbers are to be multiplied together to produce the fourth for the answer. But if the first number is more than one, the fourth number must be divided by it ; or it will be too great. Therefore, if four feet should cost but 60 cents, then the answer would be four times too much, and must be divided by 4 — ^thus : 4 : $0.60 : : 1800 : $270.00 .60 4)1080.00 270.00 Therefore, these arrangements and operations meet all possible cases, where we can say — if the first number gives, produces, pur- chases, &c., the second, what will the third give, produce, purchase, &c. The second and third being inter-multiplied, always give the answer, on the supposition that the first number contains but one of the articles under calculation. If it contains more than one, the answer must be reduced to the truth, by dividing by it. Example. If the excavation of 27 yards of earth cost $17.25, what will the excavation of 77 yards cost ? 27 : 17.25 : : 77 : 49.19 77 )1328.25(49.19 If-one yard of excavation should cost the $17.25, the 1328.25 would be the answer without being divided by 27. But as the whole 27 yards cost but $17.25, that answer would be 27 times too much, if not reduced by this divisor. 19 ROOTS AND POWERS. Any number is called a root, when considered in relation to its powers ; and its powers are estimated by the times it is multiplied by itself. As 2x2=4 is 2d power — 2 x 2x2=8 is 3d powdr — 2x 2x2x2=16 is 4th power— 2x2x2x2x2=32, 5th power, &c. There cannot be a real exhibition of any power above, the 3d, or cube. But it is often necessary in a calculation to advance, ideally, into the higher powers. No student is qualified for entering upon the study of the Mathe- matical Arts, without a good knowledge of the square root and of the cube root. These are made as familiar, in this treatise, as any diligent student can desire. Hutton's concise method for approximation in all the higher pow- ers, (sufficient for ordinary practice,) is given at the end of the roots. SQUARE ROOT. Sec. 22. A distinction between the Squdre and the Root must be understood by students. A root multiplied by itself produces the square : as the root 4 multiplied by itself, produces the square 16. When we use the word root, with or without the word square pre- fixed, we mean the root only. When we intend to express what is produced by multiplying the root by itself, we use the word square only. Operations under the head of square root are divided into Invo- lution and Evolution. It is an operation in Involution to multiply 10 by 10 and produce 100. It is an operation in Evolution, hav- ing 100 given, to find the number which multiplied by itself will produce the given hundred. The operation is called extracting the root. iNvoLirnoN OF THE Sqtjare Root. ' Sec. 23. This process requires no instruction. It is simple mul- tiplication, limited to multiplying the same number by itself; as twelve times twelve give one hundred forty.four. 20 Evolution of the Square Root. Sec. 24. This operation, when applied to squares, whose roots are found in whole numbers, within the limits of common multipli- cation tables, is a mere mental operation requiring no figures. As the root of 144 is 12— of 121 is 11— of 100 is 10— of 81 is 9— of 64 is 8— of 49 is 7— of 36 is 6— of 25 is 5— of 16 is 4— of 9 is 3— of 4 is 2 — of 1 is 1 . No intermediate case is found among these examples, which can be thus easily and mentally solved. But all the intermediate squares will give fractions of numbers in their roots. And all higher numbers present analagous difficulties, and still more complicated. Hence Evolution, or the Extraction of the Square Root, requires study and method. Sec. 25. Directions for extracting the Square Root. As the number of figures in the root will always be equal to the number of pairs in the square of it, the pairs are pointed off from right to left. If an odd figure remains at the left end of the line, this alone will give one point, and of course stands in the place of a pair. Therefore we can foresee the number of figures which the root will contain, as soon as the square is pointed off. The last pair, or point, on the left, gives a number, which always contains more than half of the whole root ; consequently that point in the square ex- ceeds, in breadth, all the rest of the area of the square. Sec. 26. Take for example 45623.0000 square chains. These chains we wish to lay out in a square piece of ground. One side of this square lot will be the root of these square chains. There- 21 fore it is a side of the lot we are in search of. the first point at the left will be 2, and this must have two figures at the right of it, besides the decimals, the nucleus square A must be 200 chains on each side. The slip B, 10— slip c, 3 — slip d, .5 — slip e, .09. As the square of e .09 d .5 A Operation. 45623.0000(213.59 4 41)56 41 w 423)1523 1269 T 4265)25400 21325 s 42709)407500 384381 -10. ■200. 213.59 is the length of one side of the lot, or the root of the given square. 23119+ N. B. The teacher must shew the student to perform similar operations frequently ; then he will be prepared to understand this illustration. CUBE ROOT. Sec. 27. The same distinction between Involution and Evolution must be understood by students under the cube root as that de- scribed under the square root. 22 Evolution of the Cube Root. Sec. 28. As the number of figures in the cube root will always be equal to the number of tripple figures in the sum given for ex- traction, the given figures are pointed off" in threes from right to left. If two or one remain, a separate point is also made of such remainder. Therefore we can foresee the nuinber of figures, which the root will contain, as soon as the given figures are point- ed off", as mentioned under the square root. The cube root of the last point on the left will exceed that of all the remaining points; as all the rest must stand on its right, and, of course, diminish ten- fold in value. (Cyphers, equalling the number of points, may be annexed to it.) This cube root will be the length (or linear extent) of the nucleus cube; upon three sides of which all that remains to be extracted is placed in layers. Therefore the operation is to be so performed, as to give the thicknesses and superfices of all outer layers, and of the fiUing-in corners. Directions for Extracting the Cube Root. Operation. Cube 75686967(423 Root. 64 5044)11686 10088 532989)1598967 ^ 1598967 Auxiliary operation carried on with the above. 1st. 4x4x4=64 square of the nucleus cube. 2d. 4X4=16 area of each face of the nucleus cube. 16x3= 48 area of the three faces, being the trial divisor. 48)116(2 thickness [last figures left ofF, 86.] 96 20 23 4-{-4-|-4=12 length of the three parallelepipeds. Length of the cube — 2 set to the right, as of less value. 122 Width same 2 244 area of all the cube parallelepipeds. Area of the faces 48 5044 true divisor for second figure. 3d. 42x42=1764x3=5292 area of the three faces. 5292)15989(3 [off 57] 42+42+42=126 length of parallelo- thickness 3 [pipeds. 1263 3 So on like the rest above, &c. Sec 29. Explanation. Students must inspect a model, while examining this calculation. If no wooden model is prepared, pro- ceed thus : Cut a potatoe cube, about one inch. Cut three potatoe slabs one fourth of an inch thick, whose area (each) will precisely cover three of the sides of the cube — pin them on their respective sides, so that when on, the whole shall be an enlarged cube ; but with unfilled corners. Fill the three long corners with potatoe parallelopipeds ; and fill the place where all the pieces leave an open corner, with a cube of the same thickness of the parallelo- pipeds. By the operation annexed the nucleus cube is obtained by trial ; that is, by multiplying the nearest figures, until the nearest is found within the point (75 in this case,) as 5x5x5=125 — this being too high, say 4x4x4=64 — this being the nearest below 75, consider 4 the meeisure of the nucleus cube. After bringing down the next point, divide by a trial divisor, made up of the three areas to be covered with slabs ; the quotient will be their thickness. The same thickness will equal the breadths of the parallelopipeds and of the corner cube. Their united length multiplied by their breadths, and added to the aforesaid areas, give the whole area. 24 Explanations. This area is the true divisor of the 11686 in this example. In adding the area of the three faces, of the parallelopi- peds, and of the corner cube, their respective areas must be arrang- ed from left to right according to their values. In this case, 48 is, truly, 4800—12 is 120—2 is unity. Therefore they add thus: 2 120 4800 4922 But the areas of the parallelepipeds and corner cubes being ob- tained by a joint multiplication, they form 244 before 48 is added. This explains that position of them. Students will perceive that the area of but one face of the paral- lelepipeds and corner cube are calculated. This is on account of their mitred character ; as the middle of each side, only, is reckon, ed in their areas. (See figure.) The lengths of the two dotted lines give the area, which is equal to the whole of one side ; there- fore one side, only, is taken. Sec. 30. Hutton'^s approximating method for the higher poioers. Make several trials, until the number for the root, above and be- low the true root, is found. Involve the number below as the root to find its cube, &c., according to the power required. Call this root an assumed root — this power an assumed power. Then say by the rule of three, as the sum of the given, and double the assumed cubes is to the sum of the assumed, and double the given cubes, so is the assumed root to the root required. The fourth power being 67543, to find the Root. 17x17 three times gives 83521. This is above. 16x16 three times gives 65536. This is then the first below. 25 First Operation. The assumed power, or the power of the assumed root 16, is 65536. 65536 assumed. 67543 given. 2 2 131072 67543 given. 135086 65536 assumed power of root 16. 198615 : 200622 :: 16 : 16.1616 Second Operation. 67038.79 2 67543 2 134077.58 67543. 135086 67038.79 assumed power of 16.164- 201620.50 : 202124.79 :: 16.16 : 16.2 TMrd Operation. 68874.75 2 67543 2 137749.5 67543. 135086 68874 assumed power of 16.2 205292.5 : 203960.75 :: 16.2 : 16.09 As far as these operations are carried on, the nearer they ap- proach the true root. But these alternate above and below the truth ; and the medium is not to be considered as the truth. When the root is supposed to be nearly found, the proof is shewn by invo- lution. But all evolutions of the powers above the cube are long and tedious. 26 TRIGONOMETRY. Sec. 31. Angles, or corners, are: 1. Right angle, square corner. 2. Obtuse angle, larger than a square corner. 3. Acute angle, smaller than a square corner. Sec. 32. Triangles, or three-sided figures : 1. Right-angled triangle, having one right angle. Its perpendi- cular line may be called the vertical leg — its bottom line may be called the horizontal leg. 2. Obtuse-angled triangle, having one obtuse angle. 3. Acute-angled triangle, having all the angles acute. 4. Isosceles triangle, having two of the sides of equal length ; consequently, having two equal angles. 5. Equilateral triangle, having all the sides of equal length ; consequently all the angles equal (just 60 degrees each.) This triangle necessarily includes the isosceles triangle. 6. Scalene triangle, having no two sides of equal length. The three first named kinds of triangle may be scalene. Sec. 33. Five miscellaneous items, which should here be noticed by the student. They will be farther illustrated. 1. A trapezoid is any four-sided figure with only two parallel sides. Any figure bounded by right lines may be cut into trape- zoids by latitude and departure lines [to be explained further on.] And the superficies of each trapezoid may be found by adding the parallel sides, multiplying their sum by their distance from each other, and halving the product. 2. Equal triangles. Two lines are parallel, when equi-distant from end to end ; and all possible triangles, made between them on the same base, contain equal superficies, or areas. Any figure bounded by right lines may be reduced to a single triangle by the application of this principle. [To be explained further on.] 3. Degrees of a circle, are 360 ; each quarter or quadrant con- taining 90 degrees. The sine of an angle is a line let fall from one end of the arc of the angle, perpendicularly upon the opposite side. As the line a c is the sine of the angle e. 27 4. Line of cJiords, is a graduated line connecting the two ends of a graduated quadrant by a chord Hne. It is made by setting one foot of the dividers at a, and extending the other foot to each degree on the arc, and turning it down to the chord line. The chord hne is used in plotting or projecting, when geometrical calculations are to be made. 5. A triangle contains 180 degrees; for ahalf circle contains 180 degrees, which is represented by the serai-circle g h i. By inspec- tion, without taking the steps of a demonstration, the reader will per- ceive that the angles a and a are equal, also c and c, also e and e. Now as the angles ace below the line g i, are measured by the semi-circle g h i, it follows that the angles a c e in the triangle above the line g i, may be measured by the same ; consequently, contains 180 degrees. The same elucidation may be given of all forms of the triangle. 6. In every right-angled triangle, the long oblique side is called the hypothenuse. The other two sides are called the legs. And it may here be shewn the student, that the square of the two legs is equal to the square of the hypothenuse. This is a very important principle in practical trigonometry. Sec. 34. Geometrical trigonometry gives all the sides and all the angles of a triangle, if two angles and one side, or two sides and one angle, are previously taken by the proper measures and observations. From the following exemplification of this proposition, a reader of ordinary ingenuity, with no previous knowledge of trigonometry, may make all the applications which the following treatise requires. Draw the lines and angles given, then finish out the triangle in the only way in which it can be completed, without any random opera- tion. The sides 70, 80, in the figure, and the angles at them are given. Draw the given line fifty feet, calling any division of a scale a foot. Strike the arc 80, d, with a radius of 60 degrees, taken with the dividers from the line of chords on Gunter's scale. Take 70 degrees, being the angle at 70, from the same line of chords, and set it off, on the arc, which will extend to d. Draw the line from 70 through d indefinitely. Then with the sweep of 60, again strike the arc 70, /, and set off 80 degrees upon it, being the angle at 80, which will extend to ef. Draw the line from 80 through / indefi- nitely, and it will cross the line which was drawn through 70, at g. 28 Where these lines intersect each other, is the true place for the other angle. Measure the two new sides by the same scale by which the given line was laid down, and you have all the sides. Add the two given angles together, which will make 150 degrees. Subtract this sum from 180, the degrees always contained in a whole triangle, and the remainder will be 30, the degrees of the new angle at g. Sec. 35. Proportions of sides and angles. The angles of all tri- angles are proportioned to their opposite sides ; therefore when two angles and a side, or two sides and one angle, are given, the other angles and sides may be found by the common rule of proportion, (rule of three.) But sines (as represented in number 3 of Sec. 33, a c) are used instead of degrees. These sines stand in tables, ac- cording to their actual lengths — calling the sine of 90, one. Sec. 36. Trigonometry is a most sublime application of Mathe- matics. By it we learn the distances and movements of celestial bodies, which are millions of miles distant from us. And by it, also, we ascertain powers and movements, essential to our daily duties and comforts. Trigonometry, after all, owes its mighty pow- ers to those laws, by which we ascertain two unknown sides of a three sided figure, by having but one side measured. The adept in mathematics knows, that the science requires that its applicants should be able to find one side from two, when the triangle has a square corner in it — ^or so that the sharpness and bluntness of a corner, made by the meeting of two sides, is necessary to most cal- culations. But these may be ranked with other principles and ope- rations ; the peculiar essentials being those just stated. Sec. 37. A corner (always called an angle in treatises,) is best measured by a circle, struck around it with compasses, (usually called dividers,) with one foot standing exactly in the angle. The piece of the circle between the lines making the corner, is the measure of the angle. Every circle has always been divided into 360 degrees. Therefore, if one quarter of the circle, or 90 degrees of it, is included between the lines, the angle is a square corner, or 90 degrees — so of any other-sized angle. The circle is preferred as a measure ; because a circular instrument for measuring has the most universal application. Students must learn the use of the compass, quadrant, sextant, &c., by actual shewing only. 29 Sec. 38. The two great principles applied to trigonometry are : 1. Application of the square root to the theorem, that in every triangle which has one square corner in it, (called a right-angled triangle,) the square of the slant-side (called hypothenuse) is equal to the square of both of the other sides — they being first squared separately and then added together. 2. Application of the rule of three to the theorem, that sides of triangles are proportioned to their opposite angles. That is, the greater the angle, the longer the opposite side, (meaning the side which does not touch the angle.) But this proportion is not direct, as 6 is to 18, so is 10 to 30, &c. But a table of sines, representing degrees, must be used instead of degrees. Students must be taught the use of the table of natural sines, by shewing only. NATURAL SINES. Sec. 39. It is not necessary to use tangents or secants in the ap- plication of trigonometry to the Mathematical Arts. Sines are suffi- cient in all cases. A table of natural sines is essential ; but artifi- cial sines and logarithms are not necessary. In some long series of calculations, particularly in working traverses of numerous courses, logarithms are a relief in the multiplications and divisions. Traverse tables, even then, are preferable. The student should always be fully instructed in trigonometry by the use of natural sines only. Afterwards, he will learn the use and application of tangents, secants, and logarithms, in two or three days, if required. Exhibition of Natural Sines. (See figure.) Take the triangle e s a. The angle near s is 30 degrees. The sine of 30° is a e. The co-sine of 30° is e x — which is equal to a s, and a s is considered and calculated as the co-sine of 30°. Now call the radius one inch, and the sine a e will be half an inch. Look in the table of natural sines, and you will see radius (sine of 90°) 1.00000, and the sine of 30°, 0.50000, and the co-sine 0.86603. On measuring, all will be found thus to agree in measure with the table of sines, calling radius one. The same will hold true with every degree, minute, and second. 30 Directions for using the Abridged Table of Natural Sines. Sec. 40. The sine of every degree and of every half degree is set down in the abridged table. In all full tables the sine of every minute of each degree ° ' is included. For such tables the student is referred to the numerous books in use ; but they are too long for this treatise. A common scale may contain this abridged table ; and it will be found sufficiently accurate for all the common cases in practical Engineering, and in the Mathematical Arts in general. One minute of a degree will be the greatest error ; and in all cases below 45 degrees it will never deviate a minute. Consequently, in all traverse cases it is sufficient, with the aid of the square root ; and a useful substitute in all cases where full tables are not at hand. Sec. 41. As only the sines of all degrees and of all half degrees are given, the minutes of any half degree must be found, as follows : Take the marginal figure as the augment (or increasing number) for each minute in the degrees following it. This added to the last preceding given sine, will give the sine of the degree and minute re- quired. Thus : The sine of 14° 47 ' is required. The sine of 14° 30' is set down as .25038 — the last sine preceding the sine re- quired. The marginal figure for the augment of all minutes of all the degrees from 10 to 17 inclusive, is 28. As 17' are to be added, 28 times 17 must be added to .25038 (the sine of 14° 30'.) 28x 17=476 — this number 28 (the augment for each minute,) being added, makes .25038+476=.25514. The true sine of 14° 47' is .25516 — but this error cannot throw the sine into either the minute above or the minute below. For tlie sine of 14° 48 ' would require 35, and the sine of 14° 46' would be deficient by 26. If seconds of a minute are required, divide the augment by 60, and the quotient will be the addition to be made to the sine. Sec 42. When the middle term in the rule-of-three operation, is in sines, the answer will, of course, be in sines. To find the true minute to the sine thus obtained, proceed as follows : Subtract from the answer the sine of the degree, or half degree, next less than the answer. Divide the remainder, thus obtained, by the marginal figure (or augment) set against the degree. The quotient will be 31 the number of minutes to be added to the said given degree or half degree. ABRIDGED TABLE OF NATURAL SINES. 3 ~ De- s. c-s c-s. s. De- 3 T, M^ De- s. c-s. C-s. s. De- SB'S „„^ „p, 21144 p S 3 to 7, 36.50X \ 21.50 \ =967.25 I ^^ G / ^ , „ „. .. ^ 13.00 H I 2 to 7, 35.55X | ^'^^^ | =604.39 ^^^^^ A. Q. R. Double areas=4750.76 Ans. 237 2 5 Sec. 96. Triangular casting reduced to a single triangle. In order to prove the accuracy of my calculation, I reduced the whole plot to a single triangle ; and then cast the contents by one multi- plication. Before exhibiting that operation, I will explain the prin- ciple with a small figure of but 5 sides, A, B, C, D, E. This method depends on the principle referred to in the 2d article in the 33d sec- tion. Having lost my brass slip on the road, I made one of a piece of tin with a pair of coarse shears ; and I will explain this instead of a better, such as I now have before me. I cut a slip of tin half 63 an inch wide and 9 inches long, perfectly straight. In the middle I drew a line lengthwise, perfectly straight and exactly parallel to its edges, as exhibited in the figure. The object of this slip was the same as the groove on the protractor ; that is, to exhibit parallels without defacing maps by marks, and to guide a foot of the dividers more accurately than could be done by a scratch on paper. The principle just referred to is here exhibited by the triangles D C B and D G B. For both stand on the same base D B, between the same parallel lines P, P, and 1, 1. Consequently contain equal areas. Therefore by extending the line A B to G, and drawing the line G D, there is just as much land added to the field by tak- ing in the area at B G. But it is not necessary to draw the parallel line P P. For if the dividers are extended from the angle C to the central line on the slip 1, 1, and then carried with the same exten- sion to the indefinitely extended line F G, and moved back and for- ward on said line, until it rests at a point (as G,) when it is found (by sweeping the other foot,) to be the nearest distance from the central line on the slip, that point will be in the parallel line P P, and at the angle of the new triangle. It will be seen, that by this operation, the angle at C is extin- guished ; leaving but four angles in the field. The angle E may be extinguished in the same way ; leaving the single triangle F G D. By the use of the slip, all embarrassing scratches and marks on the plot are avoided ; and the base and perpendicular of a single triangle only, are to be measured and multiplied by each other. Half of that product is the area of the field. Sec. 97. Choosing base lines. When a farm of many sides is to have its area calculated upon this plan, several triangles will grow out of each other in a manner more complicated. But if the slip is always laid so as to connect two corners, leaving one between, that one will be extinguished. One side must always be assumed and extended indefinitely for the base line, as F G ; every new cor- ner must be made on that line. The side chosen for the base line must not be chosen on account of its greater length ; for it will make the angles on it too acute for accuracy. But it must be so chosen, as to leave most of the plot standing upon it in its longest direction. After reducing the angles on one side of the plot, until they extend along the base line so as to make very acute angles, commence on 64 the other side of the field. But always work to the same base fine, and always begin with the angle nearest to it. Side A B 20.00 New triangle. B C 21.50 FG 49.70 C D 26.00 ' D G 42.00 D E 25.70 — E A 3^.60 ^ , 2)208.74000 10)104.37000 Scale 20 per inch.- 4 1 7.48 40 29 920 Acr. Q. Rd. 10 1 29 Sec. 98. Two base lines. It will often happen, that the angles formed on the base line will be too acute, even after working on both sides of the field. In such cases extend one of the new sides indefinitely, which touches the base line, and work to that as to the first base line. Then, when all the angles are reduced to four, extinguish whichever of the angles may be most conveniently ex- tinguished; without regard to any choice between the base line, whether the first or second one be finally retained. Sec. 99. Single triangle accurate. The advantages of this meth- od over that of casting the contents by separate triangles are mani- fest. Every step in the process is wrought by points, and one me- tallic line. Most errors in plotting arise while working to the scratch lines on paper. If the points are pricked in \/ith sharp round instruments, and the paper is old and of a firm texture, we can work to such points with more accuracy than can be expected from the most skilful survey. And a hne accurately engraved on copper, and above all on tempei'ed steel, will scarcely admit of an error. Considerable practice is necessary in this case, as in all others. Sec 100. Trapezoidal method. The third method which I adopted for proving my calculation, was the trapezoidal, or latitude and de- 65 parture method. It is constructed upon the following plan. Let a meridian line be drawn on the east or west side of the plot, so as to touch its ext^-eme side or corner, as the dotted line 4, 9, in the figure, which touches the extreme corner marked 1. Let two lines be drawn perpendicularly from this line, so as to touch the north and south extremities of the plot, as 4, 4, add 9, 9. Now calculate all the area included within the meridian line, the two perpendicular lines aforesaid, and the outside lines from the end of one of the per- pendicular lines to the other ; as 4, 5 — 5, 6 — 6, 7 — 7, 8 — 8, 9. That is, all the land, both inside of the plot and outside of it, be- tween it and the meridian line 9, 4. Then cast all the said outside part, and subtract it from the sum of both inside and outside ai'ea ; and the remainder will be the inside area, or true contents of the survey. Sec. 101. Accuracy of tlie trapezoidal method. The advantages presented in this plan are manifest on inspecting the figure. It will be seen that the whole may be cut into trapezoidal figures. Or that the north and south sides of each will be parallels, standing perpendicularly on the west line. All the sides but one, of each figure, are meridians and parallels of latitude ; consequently they may be calculated like latitude and departure in traverse sailing. Then their contents may be found by adding the departures bound- ing each trapezoid, multiplying them by their latitudinal distance from each other, and halving the product. For example, the line 5, 5, (51.48) is the north boundary, and 6, 6, (57.15) the south boundary, and the line 6, 5, (57.56) the latitudinal distance, of a trapezoid. Then 51.48 +57.15 x 57.56-:-2=3126.3714. Sec 102. In and out areas. By a little attention to the figure it will appear, that when the measure of the latitude is from north to. south, it gives the length of the trapezoids both in and out of the plot ; and when the measure of the latitude is from south to north, it gives the length of the trapezoids outside of the plot. Hence it fol- lows, that when the southern measure is the multiplier the products must be added together for the inside and outside area ; and when the northern measure is the multiplier the products must be added together for the outside area, and subtracted from the other area. When an inner angle extends backwards, as that marked 3, the area 9 66 is cast in and out twice ; but still the rule does not require any va- riation. Sec. 103. Form of arrangement. To avoid confiision, the follow- ing tabular formula was constructed. It will be understood by in- spection, after reading the preceding sections. Traverse tables, such as are used in navigation, are used for finding the latitude and departure in this kind of calculation. But it is almost as easy, and much more simple, to multiply the chains and links by the sine of the course for the latitude, and the co-sine of the course for the de- parture. Sec 104. North and south areas. In the figure the single dotted lines on the dotted side of the double lines, are the departures of northern areas. And the single lines of long dots and the long-dot sides of the double lines, are the departures of southern areas. Courses iiiid l>Hr.iiicrs I N. 1 S. 1 K. 1 VV ildfpl2dep| N. Area 1 S Area 1. iN 43 ;t-4tli.^ K 47.80 34.5,i 3305 33.0.i 33.05 1141.2165 2. S 42 3 4tlis E 18.20 13 37 12 35 45 40 78.45 1048.8765 3. N 16 W 45.20 43.45 12.45 32.95 7-35 3404.3075 4. S 77 1 4th E 19 00 4.19 18 53 51 48 84-^3 3.53.7617 5. S 5 R 57,80 57.56 5.67 57.15 ias.G3 6252 7428 6. South 16.07 i iC.07 57.15 114 30 1836 8010 7. S 63 1-2 W 19 SO 8.84 18.71 38.44 95 59 845.0156 8 West 17.02 17.02 21 42 59 86 9. N 33 1-2 E 19 00 15 84 10.49 31.91 53 33 844 7472 10. .\ 79 W 32 60 0.21 3191 00 00 31.91 198 1611 100.031 100.03 80.09 60.09 5588.4323 10337.1976 5588.4323 4748.7053 r237A cr. 1 Qr. 30 Kds. Sec. 105. Merits of the three methods. Having cast the con- tents of the farm by three methods, all of which I have long used in practice, it may be proper to express my opinion on their relative merits. I say, most decidedly, that the method of plotting and re- ducing the plot to a single triangle, is the best known method, for ordinary cases of farm surveying. For smooth even plains, and for city lots, the trapezoidal method is best. I have no room here for my reasons at length. But who will not perceive at a glance, that uneven land requires an averaging method, which is not prac- ticable by any method but by accurate plotting 1 Moore himself (the inventor of this method, of whom I learned it personally before he published it,) acknowledged that defect in his latitude and depar- ture method, as he named it. 67 Sec. 106. Heights and distances. Having completed the survey of the farm and cast its contents, it remained to calculate the dis- tance and height of the meeting house, observed while running the line 3, to 4, before I proceeded to divide it. In the field book un- der section 89, it appeared that observations were taken at two stations, o and u, which were 16,10 apart. At one station the bear- ing being N 65 W, and at the other S 51 W, it appeared by inspec- tion that the angle at o was 49 degrees, and at m 71. These sub- tracted from 180 left the angle at the meeting house 60 degrees. By section 44, the length of the line from the house to o was found thus: as .86603 (the sine of 60) : 16.10 :: .94552 (the sine of 71) : 17.57. Then I found the height of the meeting house thus: the angle of elevation being 4f , I subtracted it from 90 degrees, and it left 85| degrees for the angle at the top of the steeple. Then as .99692 (the sine of 85 J degrees) : 17.57 (the distance from the meeting house to o) :: .07846 (the sine of 4J degrees) : 1.38=91 feet. That is, the distance was 17.57, and the height 91 feet. Sec. 107. Heights and distances geometricaUy. I made a calcu- lation also, by a plot thus : I laid down the courses and distances, from the points o and u, indefinitely, and measured from o to the point of the intersection of the lines ; which gave the distance. Then laying down that distance as the base, raising an indefinite perpendicular on one end, and laying down an indefinite line from the other end at an angle with it of 4J degrees, the point of inter- section gave the height. But I laid down these plots by a much larger scale than I had used for plotting the farm. Remark. If I had made frequent trials along the line 3, 4, so as to have found the point where the beaiing of the meeting house would form a right angle with the line, both calculations might have been made by the formula 57, with and without reducing. Sec. 108. Division of land. Before making calculations for dividing the farm, I inquired whether any points were fixed upon. I was thereupon directed so to divide the farm that William should have the north end, and 10 acres less than Robert ; and that one end of the division line should be on the line 1, 2, at z, the margin of the pond, where we made the first offset. On looking over the map, [see section 81] I found that the corner 6, would be nearer to the termination of a division line, starting at z, than any other cor- 68 ner. I drew a line through z, 6, indefinitely, for a base upon each side of which a triangle was to be formed, as described in section ninety-six. By this operation I found 1287.50 on the north side, and 1086.88 on the south side, of said base line. Consequently the east end of the division line required to be moved 6.60; far enough to include half the difference, 100.31 added to half of the 10 acres=150.31. Sec. 109. Dwision hy a triangle. Here I was obliged to intro- duce a new application of the section 52, one, two, relating to the areas of parallelograms and triangles. For as a triangle is half a parallelogram, it is manifest, that if the quantity of land to be taken from the north side (5 acres and half the difference between sides) was doubled and that sum divided by the line z, 6, (found to be 45.50) the quotient would be the perpendicular of the triangle to be added; thus, 150.31x2-^45.50=6.60, the distance to which the east end of the line z, 6, was to be removed north. I drew the line z, a, and measured its course and distance on the map, which I found to be S 87^ E 44.00. Then I went to the field, run and marked said line and set up the necessary boundaries. I gave each an entire field book of his separate share, after altering the line 6, 5, to accommodate the divisions to the divided field notes. Remark 1. It must not be forgotten, nor overlooked, that after finding, that 300.62 was the difference between what was made by the assumed division line, z, 6, and what was required, but half that sum was to be taken from the north part — as one acre taken from the one part and added to the other, will make a difference of two acres between the parts. Remark 2. This method of dividing a farm, and casting the parts separately is an excellent method for proving a survey. A farm of numerous sides may be thus divided into three or four parts for more perfect proof of accuracy. It may be farther ob- served, that if we were sure that the survey and calculations were correct, a farm might bs divided by a ceJculation made on one side of the cross line only. Sec. 110. Road surveying. Road surveying, when nothing more is required than courses, distances, and notices of objects, be- longs to Agrometry. But when ascents, descents, dug-ways to be made, beds to be raised, &c., are to be calculated, it belongs to 69 Engineering, described hereafter. In road surveying, however, the field notes are kept differently. The chain is carried directly on, without starting anew at the angles or turns in the road, as in farm surveying. But the change of course is always set down, like other incidents — and the leathern tallies are carried as directed in section 77, until the seven are moved, and then another tally is run ; when the hind-bearer will be reminded that eight tallies are out, by finding no tally to slide. Then he notifies the surveyor that one mile is completed. This is entered in the field book, unless the hind-bearer is also entrusted with the mile entries. Sec. 111. Road, field look. The following is the form of the field book of a common road survey. Field look of a Road Survey from Catskill Village to Catskill Mountains. Beginning at the west end of Benton's bridge. 0.00 S 85 W. 10.21 Meiggs' house, left. 31.20 Gordon's house, I'ight. 36.07 Cross road from Catskill to Fall mills. 42.90 S 72 W. 140.31 Long Swamp. 247.60 N 64 W. 317.90 S 86 W. 560.41 Kisketam flats, at Coat's land. 640.27 Greene Patent ledge in M. Lawrence's range. Same N 79 W. 720.00 Main Catskill Mountain. in. OROMETRY. (Line surveying, reviving lost lines.) Sec. 112. Running lines from a plot. \n Orometry^ihe courses and distances are always given ; but the objects of the survey are various. Sometimes the out-bounds of large tracts of land are run and established, and the tracts are cut into lots on paper ; then the business of the surveyor is to run, and mark, the boundaries. In such cases the surveyor must be very careful to plot from level out-boundaries ; and, in cutting up the tract into lots, he must be 70 eqimlly careful. For without such care, future tenants, or pur- chasers, will have good reason to complain of uncertain boundaries. Sec. 113. Reviving old lines. But after a district of country is inhabited, the most common cases in Orometry are, the running and marking of old lines, lost by negligence. In such cases, if any one corner is remembered and can be precisely located, all the other corners and lines can be found. Much experience, however, is re- quired for searching out old lines. If several boundaries accord with each other, this accordance has great weight with honest far- mers, also with jurors, in fixing the lines. For no one will doubt the correctness of lines, when several sides of a farm coincide with the written courses and distances, even if no well established corner can be found. History of an Orometric Survey. Sec. 114. Reviving lost corners. On the 21st of June, 1803, I commenced the survey of the tract of land, called Disc's patent, in Schoharie county. New- York. It had been surveyed in the year 1743,* sixty years before I surveyed it. The boundaries were lost in most cases ; and the proprietor of Scott's Patent (which belonged to John Livingston, Esq., whose agent 1 then was) accused Mr. Rechraeyer, the proprietor of Disc's patent, with encroaching upon his tract. A survey, therefore, became necessary. But there was not a corner boundary established. Several side lines were pretty well marked. Sec. 115. Settling magnetic variation. The first point to be set- tled was, the variation of the needle from 1743 to 1803, a period of 60 years. Having been told by the Surveyor General, De Witt, by Messrs. Cockburn, Wigrai^i, Trumpbour, Van Alen, and Savage? of the State of New York; also by Samuel Moore, of Salisbury, Connecticut, that the north point of the needle had been appx'oaching the meridian for a century, at the rate of nearly one degree in twenty years, I adopted that fact for my rule of calculation. Sec. 116. Proving tlie variation. In order to make the allow- ance, without the possibility of mistake, I concluded to run thelong- * I surveyed this tract at Uiat time, (June 21st, 1803,) but it is too large to present the whole survey here. Theretore I leave out several sides, and I leave out some precise dates of the patent also. 71 est boundary line first, on which any marked trees could be found. Not being able to find a corner, I set down the compass at a place on the line where I found several marked trees nearly in a range apparently agreeing in direction with the given west line ; which was S 2|- E. I let the needle settle on the given course and then turned the north point east thi-ee degrees with my finger; as it had moved east three degrees in sixty years; according to the opinions of our most experienced observers. This brought the north point half a degree east of north ; consequently the course to be run was S ^ W. Then I turned the sights (after replacing the glass cover) so that the south point of the needle rested ^ west. I run on that course by directing the flag-man as usual, until I became satisfied that this was probably the true line, from the number of marked trees which coincided with it. Sec. 117. Proving old marks. Lest thesa should be spurious marks, I directed the ax-man to cut in above and below several of them until dead wood appeared ; and then to split out blocks at the depth of the dead wood. Here we found, more or less distinct, J. D. (for John Dise) 1743. On the outside we found a distorted D on several trees. On counting the grains (concentric cylinders) we found from 40 to 50 very distinct, in several cases ; and indis- tinct ones, which might supply the remainder of the 60 required to answer to the 60 years. As all the new layers of wood are intro- duced between the bark and wood in the form of mucilage (cam- bium) which hardens into thin concentric layers, all the grains ap- pear as if no mark had ever been made on the tree ; excepting those near the original interior mark, and the distorted outside mark. Sec. 118. Finding old corners. As no corner could be found at any place which could be relied on, and as this line and the north line run through ancient forests where numerous marked trees remained, I concluded to run these lines until they should cross each other; and then to assume the crossing for a starting corner. By tracing all the hues from that corner, if I found a coincidence with similarly marked trees on several sides, I should believe I had run the old lines truly. On making the trial, I suc- ceeded to the satisfaction of all parties concerned. But the allow- ance of one degree for 20 years was certainly too much by several minutes. 72 Remark. At the present time (1837) it is well known that the needle was stationary about the beginning of this century in this district ; and that it is now on the retrograde. Sec. 119. Aberrations of needle often imaginary. In Catskill' Greene county, New York, I run the boundary lines between two tracts called Row Patent and Greene Patent, in the spring of 1808. I could not close the Row Patent by four chains ; though I plotted it several times with particular care. I concluded to resort to a survey, with a view to ascertain the side on which the error was committed. By comparing the descriptions, finding corners with cross lines, as in the Dise's Patent, I was able to settle with satisfac- tory proof, that the fore-bearer must have lost 4 stakes, which the hind-bearar omitted to correct. As the survey had been made 70 years before, none of the assistants could be found ; though all their names were found in the field book of the surveyor, George Met- calf, Esq. While running one of these Hnes my needle varied ; and I was obliged to send home for another compass which traversed well. It was at this time, I first observed, that my needle varied in that compass always when the sights were directed nearly as on that course. This led to the discovery, that most, if not all, the de- viations of needles, which are ascribed to the attraction of iron mines, are caused by fine grains of iron, left in the card or limbs of the compass.* IV. UDROMETRY. {Marsh and aquatic surveying.) History of an Udrometric survey. Sec 120. River and harbor survey. I was employed to survey a trunk of Fludson river between Albany and Rensselaer counties, for the purpose of ascertaining which side of a middle ground ought to be selected for the purpose of improving the best channel. On the east side the channel was known to be the deepest ; but the bot- tom was rocky and difficult to excavate. But the west side was loose gravel, easily excavated. * See Sillinian's Journal, vol. 12, p. 14, where I published several facts relating to that subject ; and proposed that the needles should be very sharp and capped with brass or silver. By this means the steel point of a needle may be kept at a little distance from the limb, and defended from rust at the sharp point. It is found to be perfect in practice. 73 Sec. 121. StaJdng out the ground. First I caused my assistants to set stakes at all the turns on both shores of the river, and at all the turns on the middle ground. Then I took soundings along both channels. Wherever I measured the depth, I set up a stake by tying a stone to one end by a very short rope, for an anchor. On the other end I marked the depth, and tied colored rags to it, that I might readily see it from shore. The spots on the river-plot repre- sent the stakes, and the figures their various depths in feet. Sec. 122. Base lines. I run two base lines on shore, where I could find the best ground ; to wit p, s, and iv, t. On these lines I made marks, at p, q, r, s, t, ?/, v, and w. I chose these places so that I could see all the stakes, each at two stations ; at both of which a compass was set. Sec. 123. Taking hearings. Setting the compass at _p, I directed the sights to 21, 5, z, 18, e, and 12 ; and I noted down each bearing, numbering them from north to south. Removing to q, I took the bearing of all the stakes which were nearest to this station. In this manner I continued until I had taken the bearings of all the stations on both sides of the river — each from two stations, by one or two compasses. Sec. 124. Points of intersection. It is manifest, that (as the course of the base lines had been taken, and the distances from station to station measured,) the distance of the stakes from each other, and from the base lines, will be indicated by the intersection of lines drawn according to their bearings. And as numerous bearings were taken at each station, the plotting is easily performed ; for all the bearings from one station may be marked off without taking up the protractor. Sec. 125. Lines of the shores. By measuring at right angles from the base lines to the staked turns on the shore, or to any other accessible object, they may be laid down by erecting perpendiculars of the measured lengths, at the places noted on the' base lines. These places may be connected so as to give the true form of the shore, &c. Sec. 126. Notations on the map. The position of the stakes, with the depth of the water at each, was marked on the plot or map, at the intersections, as shown on the map. Notes relating to the 10 74 bottom, were made in the field notes, with references to the stake as marked down. Remark. AH harbor surveys may be made upon this general plan. Also any surveys of ponds, marshes, &c. But particular cases require a judicious plan, adapted to its pecuUar circumstances; and no set of rules can apply in all cases. The surveyor's judg- ment will always give character to such surveys. STATICS AND DYNAMICS ; As far as they are necessary to Civil Engineering. Sec. 127. Statics.* The science of gravitation or pressure, while bodies are restrained from motion ; as the mechanical powers when in equilibrio, or the compression of metals or timber, under heavy weights. The construction of bridges, piers, roofs, dams, flumes, &c., requires to be designed according to the laws of statics. Sec 128. Dynamics.! The science of motion, or moving force, applied or applicable to bodies when free to receive motion. As the power applied to a lever while raising a weight — the power of gravitation in giving motion to a carriage down an inclined plane — the power applied by means of a pulley in raising casks, &c. When Statics and Dynamics are applied to water, they are deno- minated Hydrostatics'^ and Hydrodynamics.^ Sec 129. Hydrostatics, applies to water, while its pressure is resisted ; as the strength of planks and their fastenings on the frame- work of flumes, must be in proportion to the square root of the head of water. Sec 130. Hydrodynamics, applies to water in motion ; as the velocity of water in a raceway will be in proportion to the inchna- tion of the plane of its descent. The various instruments and struc- tures employed in hydrodynamics, are called Hydraulics. * Statos, Greek, (from istemi,) standing, being stationary. t Dunamis, Greek, power, efficiency. t Udor, Greek, water, prefixed to States. § Udor, Greek, water, prefixed to Dunamis. ^ 75 FALLING BODIES. Sec. 131. All bodies, both solid and fluid, are accelerated equally by the attraction of gravitation. This is demonstrated by what is called the Droppers experiment with the air pump. The droppers receiver is a glass cylmder, usually about 18 inches in height. The air being exhausted, a feather and a piece of lead are dropped from top to bottom, by means of an appropriate apparatus. As the fea- ther reaches the bottom as soon as the lead, it follows, that it is the atmospheric air only, that causes the difference of velocity in all common cases of falling bodies. Sec 132. After numerous and extremely exact trials, it is found that all bodies fall 16.2 feet in one second, in a vacuum. Also, of cou S8, that in falling one foot, a body acquires such an increased velocity, that should its increase be suspended at the end of the foot, it would thence forward move at the rate of 8.1 feet per second. But the continued increase carries it 16.2 feet in one second from its starting. This principle will be more clearly illustrated with water-pressure. WATER, As AN Agent in Engineeeing. Sec. 133. Students should learn, from trial, to estimate water with facility, by weight, cubic measure, and common liquid mea- sures. A pint of pure water weighs one pound.* The weights and measures adopted in this treatise, will be 2000 Jfes. neat weight for a ton, in accordance with the revised laws of New York. Also a cubic foot of water to 60 ifes., and 28.8 cubic inches to a pound or pint of water. Let every student measure and weigh water by using common pails, cups, tubs, &c. Let vessels of all forms be measured by the inch as an exercise ; and the correctness of the measures proved by weight and a sealed liquid measure. * It is uncertain which was first adopted, the pint measure or pound weight ; but it is evi- dent that accident did not cause their agreement. 76 Sec. 134. Hijdrometers should be immersed in water also ; par- ticularly Baume's Areometer. Manner of using Baume's Glass Areometer* in ascertaining the specific gravity of liquids. In constructing this instrument two stationary points are assumed ; . and if you have none at hand, these points may be found as follows. Take a slender glass tube, with a hollow bulb at the bottom. Put into the bulb mercury or fine shot, until you sink it in pure water almost to the top. Mark the zero point at the surface of the water. Then weigh 85 parts of water and 15 parts of table salt (muriate of soda.) After the salt is perfectly dissolved in the water, bring the temperature to 57° of Fah. Immerse the tube in this solution, and mark the point at the surface of the water, for the lower termina- tion of 15 degrees. Being equally divided into 15 parts, these parts may be assumed as standard measures for any series of tubes (one ending where another begins,) for taking the relative specific gra- vities of liquids from the heaviest sulphuric acid to the lightest ether. Water used in taking specific gravity of Solids. Sec. 135. To be familiar with taking the specific gravities of ma- terials for construction, is often of great use to persons in all other situations in life, as well as to engineers. Illustration. Tie a strong silk thread or silk twine around a piece of marble weighing seven or eight pounds. Weigh if carefully, using balancing quarter-ounce or half quarter-ounce weights ; so as to bring it to an even ounce weight on the steelyard bar. Then weigh it in water, sinking it so as to be wholly about half an inch below the surface of the water. Next subtract its weight in water from its weight in air — take this remainder for a divisor, and its weight in air for a dividend ; and the quotient will be its true specific gra- vity. As the weight in air is 8 ife. 7^ oz, ; weight in water 5 ife. ^ oz. * Mraios, Greek, slender or delicate ; and metron, a measure. 77 In air, 8 ife. 7^ oz. = ft. 8.453 In water, 5 ife. i oz. = ife. 5.016 Rem. 3 ife. 7 oz. = ife. 3.437 .3.437)8.453(2.459 spe. grav. ; that is twice and 6.874 about ^Yo heavier than an equal bulk of water. 15790 13748 20420 17185 32350 30933 In this manner the solidity of materials for construction may be readily obtained — and it is preferable to the usual practice with grain- weights, for coarse materials. Hydrostatics. Sec. 136. Make a cylindrical bellows, by cutting two circles of thick board 10 inches in diameter, and nailing to the outside rim of each, with broad headed tacks, a hollow cylinder of leather. When finished it will present a leathern cylinder of strong calfskin, 10 inches in diameter and 8 inches long. Set in the middle of the top board a leaden tube of about the fourth of an inch in calibre, and 3 or 4 feet high. Let the top fit into a glass tube, 5 to 10 inches long, by a bandage of tow. When used, the leather needs to have been soaked in water several hours. Fill the cylinder with water through a plug-hole in the top board. Lay a weight on the top board, or let a student of suitable size stand on it, so that the water may rise into the glass tube. On measuring the height to which the water rises in the glass tube from the top board, and making the proper calcu- lation, this result will be found : the weight set on will precisely equal the weight of a cylinder of water, 10 inches in diameter, of the height of the water in the tube. Hence it follows, that water presses according to its height ; not according to its quantity by measure or weight. Therefore were it not for the impossibility of 78 maintaining the perpetual supply of water, a tube of an inch calibre would be sufficient for moving the machinery of an extensive factory, under a hundred feet head, supplied from a small reservoir or tub. Hydrodynamics. Sec.. 137. After this is demonstrated, that water pressure depends on its height, and the weight of a standard measure of water is ascertained, we must determine by trial, what measured velocity will be given to water by a measured head. It was before stated, that trial has shown that a measured pint of pure water weighs a pound. Sec 138. Trial has also shown, that under one foot pressure,- water will be forced through a lateral aperture with a velocity of 8 feet and a tenth, per second, in a vacuum — probably it will be cor- rect in practice to say 8 feet per second in the atmosphere, at the . precise point of effusion. This trial prepares us for the universal ~ rule which governs in all cases of motion by gravitation. Sec. 139. The increased velocity of water effused, and of falling solids, is as the square root of the head of water, and as the square root of the distance through which solids fall. Taking 8 feet of late- ral effi;ision per second for the first foot, and the increase from that zero (if it may be so called,) as the increase of the square root of the head, and the increase of the distance fallen through in the case of solids, we arrive at results of vast importance in engineering. Sec 140. Illustration. A spacious flume has 25 feet of water in depth, with five apertures or gate-holes, of one square foot each. The centre of the first gate-hole is one foot below the top of the water — the second 4 feet — the third 9 feet — the fourth 16 feet — the fifth 25 feet. The square root of one is one, and the lateral effusion of water is 8 feet per second, as demonstrated by trial. The square root of 4 is 2, and the lateral effusion of water is 8x2=16. The square root of 9 is 3, and the lateral effusion of water is 8x3=:24. The square root of 16 is 4, and the lateral effusion of water 8 x 4= 32. The square root of 25 is 5, and the lateral efflision of water is 8x5=40. All intermediate heights of head may be calculated in the same manner. That is, extract the root of the height given, in feet and decimals of feet. Multiply that root by 8, the velocity of the first second of pressure. See the annexed diagram. 79 ^ ■^ s o CO T-i 00 o o l-H II 00 X II OD X CO II 00 X II 00 X Pi w >. fo be 5b ^ J3 >-> > ^ 81 and terminated by a few inches of glass tube. Fill it with water* and invert it in a vessel of water. The weight of the atmosphere will press upon the surface of the water in the vessel every where alike ; but finding no resistance in the tube, (the air having been dis- placed by filling with water before inverting it,) the water is raised between 30 and 34 feet, until it counterbalances the weight of the atmosphere. A cheaper and more convenient method is, to use a torricellian tube, or (which is the same thing) a barometer. The height to which mercury, is raised in the tube by atmospheric pres- sure, may be readily applied to water by calculation ; reckoning mercury as 13J times as heavy as water. If the mercury rises 29.92 inches, multiply this by 13.5; it gives 403.92 inches — divided by 12, it gives 33.74 feet. This calculation is important in fixing the upper valve of a pump, in regulating water works, 6z;c. Sec. 143. Illustration. A pump maker had practised setting up pumps near the tide-water level on Hudson river. He was em- ployed by a potash manufacturer in Castleton, Vermont, to erect a pump. He adopted his accustomed rule for fixing the upper valve. On a warm damp misty day (soon after the pump was set up) it was found that water could not be raised. The lecturer to the Medical class on Natural Philosophy, was called on by the proprietor (who was a trustee of the institution,) for an explanation of this strange phenomenon. It was found, that when the air was exceedingly Ught, by warmth and by being surcharged with vapor, mercury would not rise in the torricellian tube to a sufficient height to carry, by calculation, the water to a sufficient distance above the upper valve to give play to the piston. Lengthening the piston-rod and lowering the upper valve, immediately corrected the embarrass- inent. Sec. 144. Illustration. I have been told that the head of the wa- ter works at Hudson, N. York, (two miles out of town,) was more than 30 feet lower than a high ground, over which the pipes were laid. I have often seen the head of the works in the present cen- tury, which were in good order ; but this is said of their commence- ment some 40 or 50 years ago. True or false, the principle may be illustrated by supposing it true. It is said, that in damp warm weather, the water would not run. This is in accordance with the laws of pressure. Atmospheric pressure will carry Avater, in close 11 82 air-tight pipes, over hills from 30 to 34 feet above the spring head, near tide-water level, as may be demonstrated to students, by the barometer or torricellian tube before described. GONATOUS FORCES, APPLIED TO BRIDGES AND OTHER FRABIE WORK, AND TO STONE AND BRICK ARCHES. Sec. 145. Gonatous forces (from gome, gonatos, Greek, angular flexions, like the knee joint,) are applied by means of angular flexions of ropes or bars ; as the genicular braces of carriage calash-tops, or the sailor's funicular advantage in hoisting a sail by springing the halliards out from the mast. The lavi^s of pressure applied to bridges, arches, &c., are better explained on the gonatous principle than by any other means, as follows : The weight of a bridge may be made to res!: on a point; and that point to rest on a single pillar of stone or wood. The pressure would then be downwards, and single in direction. A ton (2000 ife.) would press directly on the earth at the foot of the post. But if it pressed on the meeting point of a pair of rafters, here would be a resolution of force into two equal- parts ; one half in the direction of each rafter. And the pressure being in the line of the woody fibre of the rafters, it would not crush them without very great weight. This pressure would not be directed downwards, as if on a pillar; but would pi-ess outwards as well as downwards, and tend to spread or separate whatever supported the rafters at the foot. If the angle at the meeting of the rafters should be 126° 52', the spreading pressure would be double the downward pressure, according to the law of diagonal lines. For each rafter would be the diagonal of a parallelogram, twice as long horizon- tally, as deep vertically. This is the common method of estimat- ing roofs, bridges, &c., also arches, by estimating short chord lines, separately taken. But the same results may be produced by the application of the gonatus principle, called the genicular and funicu- lar power ; and it is more simple and of extensive generality. It will be understood by a reference to the knee braces of chaise tops, or better by inspecting the printing press, to which this power is very advantageously applied. 83 Sec. 146. As the genicular* miAfunic\dar'\ powers are very im- portant in their practical application, let students make the foUovvuig trials. Fasten two pullies against a wall about 8 feet apart. (If you have no pullies, smooth wooden pins, half an inch in diameter and five or six inches long, may be substituted.) Draw a very flexible cord or rope over them, and attach weights to their pen- dant ends. Let the pendant ends hang down from each pin about 4 feet. Now apply different weights to the middle, and to the ends of the rope. . By thus flexing the rope in the middle, this principle will be demonstrated. As half the weight applied to the middle of the rope is to one of the end weights, so will be the descent of the middle of the rope, to the length of the section of it between the mid- dle weight and pulley. It will be perceived, that this is the law of the inclined plane, inverted. Sec. 147. Students can easily be made to realize the almost uni- versal application of the law to be deduced from this experiment. For though the flexion of the rope is downward, the principle is the same. At the commencement of the flexion of the rope, one pound will raise hundreds. But let the flexion continue to be increased, until the angle is reduced to 90°, and the farther flexion will require great weight. So if the rope was changed to a jointed flexed bar of iron, and that tui'ned with the angle upward, it would resist great weight, if placed upon the outer point of the angle. The law of the inclined plane would perpetually apply. That is, the weight would press in a perpendicular direction proportioned to its pressure in a lateral direction, as the distance of the angle from a straight line with the ends of the bar, to the length of a side of it. Sec. 148. The application of this law to frame- work or to stone arches, may be farther illustrated by setting the pullies or pins in the wall or ceiling, so that one shall be 4, 5, or 6 feet higher than the other. Then fasten the middle weight to the middle of the rope by a piece of twine. Here all the weights will hang down, present- ing a fair exhibition of the action of the law of gravitation. And the same law will apply to the flexion from a straight line compared with the length of the line between the middle and the pulley. The student should be taught to plot arches in short chord lines ; and * Genicular, from genu, Latin, a knee, t Funicular, from funis, a rope. 84 then to lay down each according to the laws of the funicular power ; transferred to the genicular, as above described. Sec. 149. An exhibition of the application of the genicular power to arches, &c., may be cheaply made, as follows : Get out about 12 strips of wood, like common rulers, about an inch and a quarter in width, one third of an inch in thickness, and a foot in length. Let these be united in parallel pairs, and the six pairs be joined by a two inch piece, forming a free double joint with screw rivets. This six feet of six free joints, may be bent into every form of arch ; and, by tightening the joints with the screws, the arches may be made to exhibit all the various effects of pressure. By hanging common cast-iron weights on various parts of the various arches into which the joints may be bent, every view may be reduced to mathematical calculations. MECHANICAL POWERS. Sec. 150. The mechanical powers are, elementarily, but two — the Lever and the Inclined Plane. Lever is subdivided into Lever proper, Wheel and Axle, and Fully. Inclined Plane is subdivided into Inclined Plane proper, Wedge, and Screw. Sec. 151. Lever is prying, when the fulcrum is between the power and weight. Lever is lifing, when the weight is between the power and ful- crum. Lever is radial, when the power is between the weight and ful- crum. Wheel and Axle is a perpetual lever, either radial or prying. PuLLY is a perpetual lifting lever. In all levers the power is to the weight inversely as the side to which 'it is applied is to the side applied to the weight — the bar, or its equivalent, being balanced. Sec. 152. Inclined Plane Proper. In all inclined planes the power is to the weight as the height of the elevated end is to the length of the plane. Wedge. The power is to the resistance as the thickness of the head, to the sum ^of the length of the two slant sides. But the use- fulness of the wedge does not depend much on its advantages as a 85 power, when applied to splitting rails, &c. But it gives direction to a very great degree of momentum, acquired by swinging a beetle, sledge, &c., and suddenly expending all of it on the head of the wedge. Screw. The power is to the resistance as the distance between the threads of the screw is to the length of a circular thread. The screw is chiefly useful in giving a very advantageous direction to the lever, as in case of the cider-press. It is also useful in making delicate mathematical instruments ; as an index on its head, or handle, may indicate the smallest possible degree of movement. Remark, These general principles are sufficient ; for students must see the instruments and use them. It is impossible rightly to understand them from mere description. Olmsted's Compendium of Natural Philosophy may be profitably consulted on the subject of these powers. ARCHITECTURE. Sec. 153. Though Architecture is made up of artificial mate- rials, it is truly a science. The historical origin of some of the ele- mentary principles is rather obscure ; but in most cases their history is known. The early ornamental buildings were of stone and brick. And when wooden buildings came into use, the wood was made to imitate stone in general form, sculpture, and painting. Hence it was that pillars became the elements of Architectui'e, whether of stone, brick, or wood. It has been suggested that the Gothic style grew out of the ancient practice among the Goths, Vandals, and other northern barbarians, of binding together the tops of slender trees, for covering with skins, bark, &c. That this gave rise to the sharp arch, and other peculiarities pertaining to that order. Sec. 154. The pillar consists of the hase, shaft, capital, architrave, frieze, and cornice. A column includes the base, shaft and capital. The pillar is extended laterally, or rather spread out, so as to con- stitute walls, bases, capitals, cornices, &c. And when the form of a column is assumed, its proportions are extended to all parts of a regularly constructed building. And this principle is rigidly applied to bridges, ornamented boats, carriages, rail-road cars, &c. Sec. 155. As pillars are the elements of the science of Architec- ture, they are to be first presented to the student. Figures or draw- 86 ings of the elementary pillars will greatly aid the student ; therefore he is referred to the various treatises on that subject. Benjamin's Practical House Carpenter is the most popular work in this country ; and, perhaps, quite as useful to the student as any other work. But the most efficient method of giving instruction on this subject is, to take students about a city or village ; and first, point out the va- rious orders of architecture by correctly constructed pillars — second, point out the errors ; which are always abundant — third, point out the lateral extension of pillars and parts of pillars, in the construction of walls, ceiling, door casings, window casings, cSzc. The teacher should call his pupil's attention most particularly to bridges, and other works, which do not come particularly under the daily opera- tions of common carpenters. THE FOUR PILLARS. Sec. 156. The four pillars are called Tuscan, Boric, Ionic, and Corinthian. The Composite order is of modern application. It is a fanciful intermixture of the elementary characteristics of some of the four orders — generally of the Ionic and Corinthian, This trea- tise being a mere practical text-book, will include the essentials of students' recitations, only. TUSCAN ORDER. Sec. 157. This order is the plainest and stoutest of all the orders. But it varies in proportion to the weight it is to sustain. In some of the cases of best appearance, the column is in height equal to seven times its diameter adjoining the base ; and the entablature is two diameters. Its parts may be thus enumerated : a square plinth be- low the base ; a base of a large moulding of a half cylinder ; a terete, or tapering cylindrical column ; a plain capital at the top of the column ; a plain architrave sitting somewhat towards the inside of the capital ; a frieze on the architrave ; and a broad over-laid coraice, projecting considerably forward — the entablature is thus made very plain. DORIC ORDER. Sec. 158. This is also a plain order, and resembles the Tuscan. 87 A little more slender than the Tuscan, and more ornamented. The columns are often fluted ; and the entablature and cornice are often ornamented with triglyphs and modlllions. IONIC ORDER. Sec. 159. This order is always distinguished by the volute or scroll. It is more slender than the Doric order, and is often highly ornamented with various kinds of sculpture. This order is more adopted in this country than any other, in all buildings of taste. The columns are mostly fluted ; and the architrave and frieze are more or less ornamented. The base is surrounded with more compli- cated mouldings than the Tuscan or Doric ; and the modillions are larger and generally of good workmanship. CORINTHIAN ORDER. Sec. 160. This order is distinguished by its plumose capital, and its more slender and delicate proportions. The base generally re- sembles that of the Ionic, and its column is generally fluted. Its architrave, freize, and cornice, are ornamented with sculptured work and curvelinear modillions. But its high elegant capital gives it a degree of grace and beauty, far exceeding the other orders. COMPOSITE ORDER. Sec. 161. This order, though compounded of two others, has its true characteristics. It has the plumose capital of the Corinthian order below, and the volute of the Ionic order above — but the volute is generally elliptical in a vertical direction. Its architrave and frieze are often highly ornamented. PEDESTALS. Sec 162. These are proportioned to their respective orders. They consist of hase, die (parallelepiped trunk,) and a cornice — altogether about a third as long as the column supported by them. PILASTER. Sec. 163. Pilaster is a pillar of any of the orders, which has the appearance of being partly sunken into the wall — or it may be de- 88 fined, as a pillar with a thin tapering parallelepiped column. It consists of a plank, or slab of free-stone, attached to a wall, fire- jambs, &c., constructed upon the principle of some of the orders. MISCELLANEOUS STRUCTURES. Sec. 164. Colonade, a series of columns, which ai-e united by en- tabliture at their tops. Arcade, several receding arches in succession, penetrating into, or through a building. Balustrade, a series of small pillars, as those supporting stair- railings, &c. A kind of parapet. Attic, often used for a garret ; because the pedestal-like pillars, which hide the roof (garret room) are called attics. Parapet or Battlement, any low wall or balustrade, surrounding a roof or covering of any works ; intended to conceal an unseemly part, or to defend it. Sometimes it surrounds roofs for the purpose of supporting the hand-railing of a walk. Pediment, generally a small triangular front roof, set in upon the slope side of a larger roof. Balcony, open gallery; as the iron galleries around upper win- dows, and highly elevated terraces. Belvidere, (beautiful view. French.) observatory, and turret. Cupola, a dome, or smallish room, on the top of a building ; as a belfry, or hemispheric sky-light. Terrace, elevated walk. Coping, top or binding-stone, or binding timber, on a wall. Saloon, a vaulted spacious hall. Corridor, a large hall or passage to distant apartments. Some- times applied to galleries or covered ways, leading around build- ings. Lintel, and Threshold, top and bottom pieces of a door opening — sometimes applied to the top, or covered part, of a projecting out- side room. Niche, appHed to recesses in walls, for setting ua urns, dz;c. 89 RAIL-ROADS, &c. Surveying a Route for a Rail-Road, MAdam Road, Turnpike Road, Canal, SfC. EXTEMPOKANEOTTS TrAVEESE. Sec. 165. If a road of any kind, or a canal, is proposed, across a mountainous, hilly, or even a moderately uneven country, a kind of extemporaneous traverse should first be taken. This will give the directors such a general profile, or rather view of the country, at a small expense, as may enable them to judge with considerable accu- racy, respecting the most expedient route for a preliminary survey. It may be conducted advantageously with a compass, chain, and barometer. Sec 166. Let the barometer be set up at the beginning of the line. Let the flag remain with the barometer, while the compass and chain are carried to the first considerable elevation. or depres- sion, within view of the flag. Here run a line, of a suitable length, between two stakes, for\he base line of two triangles — the apex of the first triangle to be at the first station of the barometer, and the apex of the other triangle to be at the next station of the barometer. The base line being carefully measured, and the angles at each end, formed by the line with the bearings of the barometer at each sta- tion, will be all that is necessary for taking two strides with suffi. cient accuracy. As the barometer will give a good approximation to the true height at every station, and as the distances may be pretty accurately taken ; a profile across an extensive district may be taken in a short time. From five to ten miles per day may be taken by an experienced surveyor, and a skilful barometer-bearer. If notes are extensively taken, much of the character of the country maybe presented on, or accompanying, the profile and sketch-book. Sec. 167. The barometer should be applied twice at every sta- tion in the same day. It is found, that when the atmosphere is uni- form, and the barometer is not influenced by change of temperature nor moisture, the barometer will sink the lowest at about 4 o'clock in both forenoon and afternoon ; and will rise the highest at 9 in the forenoon and 1 1 in the evening. This is supposed to be caused by an uniformly operating cause, above clouds or other modifications of aqueous vapor. Therefore the barometer ought to be twice set on 12 90 the same day and place ; so that where it stands at 4 o'clock in the forenoon or afternoon, it should again stand at 9 in the forenoon or 11 in the evening, and an average be made. But when this cannot be conveniently done, it may in some measure be approached, by assuming opposite times nearly in contrast ; as 9 A. M. and 4 P. M. But the medium hours do not need contrasting ; as 7 A. M. or IIP. M. But all cases require, that morning vapors should be exhaled before the barometer is used ; and that the season of the year should be chosen, when the weather is most dry and settled. Sec. 168. Formula for calculating Heights by the Barometer, ac- cording to Hulton. As the density of the atmosphere, consequently its weight, dimi- nishes in a geometrical ratio of its height, and as logarithms of num- bers are constructed upon the same principle, Hutton sought, with success, a formula for applying logarithms for taking heights of hills, mountains, &c., with the barometer. He found that, if the tempe- rature of the atmosphere stood at 31 degrees of Fahrenheit, the dif- ference in the first four figures of the logarithm, for every hundredth of an inch on the barometer between the bottom and top of a hill, gave just one fathom (6 feet.) Hence his rule: If the mercury in Fahrenheit's thermometer (alwa3's attached to the barometer) .stan^^s at 31°, take the height of the mercury in the barometer, in inches and hundredths of inches, at the top and bottom of the hill. Find the logarithm of both. Subtract the logarithm of the top from the logarithm of the bottom of the hill. The four first places of figures in the remainder are fathoms, and the remaining ones are decimals of fathoms. If the answer is required in feet, multiply the fathoms and decimals by 6, and the product will be the answer in feet and decimals of feet. Sec. 169. If the thermometer stands above 31°, an addition will be required ; to be produced by the following formula : Divide the fathoms and decimals of fathoms, by the constant number, 435; and multiply the quotient by the difference between 31° and the number of degrees on the thermometer. This product is to be added if the temperature is above 31°, but to be subtracted if below. Note. If there is any difference between the temperature at the top and bottom of the hill, the average is to be taken. 91 A portion of a table of logarithms, sufficient for our highest moun- tains, is inserted at the end of this treatise. Sec. 170. Gregory's formula has the advantage of being conve- nient for the memory in the absence of tables. A perpetual formula of 55 with three ciphers — 55 degrees of temperature as the stand- ard — 44 with a cipher, for a corrective for deviations from said standard. His rule is : Divide the difference between the top and hottoin hundredths of an inch, hy the sum of both heights; then multiply the quotient by 55000, and the product will be the height in feet. But if the temperature differs from 55°, add the 440th part of the height for every degree it exceeds 55°; and subtract the same for every degree below 55°. Example : Barometer. Thermometer. "Top of hill, 30.00 Average, Bottom, 29.80 68°— 55°= 13 Difference, .20 Sum, 59.80 59.80).200000(.00334 .00334 17940 55000 formula. 1670000 1670 440)183.70000(.04175 1760 13 20600 17940 26600 23920 2680 770 440 3300 3080 .12525 .4175 .54275 183.70000 2200 184.24+Answer. 2200 92 Latitude and Longitude Surveys. Sec. 171. Very extensive RaiJ-roads (like that now in progress from Lake Erie to New-York,) or Canals (as that from Lake Erie to Hudson river,) should have all remarkable points, along the va- rious proposed routes, accurately settled by their latitude and longi- tude. This would greatly aid the judgement of directors; and greatly benefit large districts of country, by furnishing established points for future reference. Sec. 172. Latitude is most conveniently taken by the sextant, at noon. But Longitude ought to be taken, inland, in most cases, by the eclipses of Jupiter's satellites. I will, however, describe the me- thods of taking longitude by Jwpiter^s eclipses and by three-hour lunar ohservaiions. Sec. 173. Taking latitude at noon with the sextant, requires a nautical almanac; though some of the larger almanacs of the common kind, contain the sun's declination, and may be used as a substitute. But no engineer should fail to provide himself with the Annual Nautical Almanac ; always published three years ahead, by the Messrs. Blunts, of New-York. Note. Mr. Gates, of Troy, will furnish them to order. Sec. 174. In taking the latitude at noon, a reflector is necessary. Bowditch prefers a bowl of molasses to a glass reflector. Set a bowl of molasses (a large soup-plate is preferable,) on the ground, and take the angle between thw sun and its reflected image in the molasses, by bringing them centre to centre. This gives the double altitude ; as the distance of the sun's image, below the level of the surface of the molasses, is equal to that of the real sun above it. Halve this double altitude, which gives the true altitude. Sec 175. Having obtained the sun's true altitude, proceed to cal- culate the latitude, as follows : Look out the sun's declination — If the declination is north, (as it must be, from the 21st of March to the 21st of September,) subtract it from the altitude ; and then sub- tract that remainder from 90 degrees, which will leave the latitude. If the declination is south, (as it must be, from the 21st of September to the 21st of March,) add it to the altitude, and subtract the sum from 90 degrees, which will leave the latitude. In short days, wher^ 93 tthe sun runs low, an allowance may be made for refraction, accord- ing to the table of refraction at the end of this treatise. Note. In the longest days of summer, the sun will be too high at noon to admit of double altitude within the range of the sextant. Several methods are in use for obviating this difficulty. The fol- lowing method may be adopted : 1st. Take the double altitude of the ridge of a house-roof, or some other straight horizontal line. Then wait for noon, and take the single altitude of the sun above said ridge, &c., and add it to half the double altitude of said ridge. Sec. 176. In taking the lojigitude by the eclipses of Jupiter^s satellites, no instruments are necessary but a telescope and a good time-piece. On land this method of taking longitude is the best. Proceed as follows : Look in the Nautical Almanac, in the monthly table of Jupiter's satellites, and find the time of the nearest immer- sion or emersion eclipse. Be prepared with the true time and teles- cope. Direct the telescope to Jupiter, with the slide drawn so as to give its largest size, a few minutes before the time of the eclipse. With the eye on Jupiter, move the slide so as to diminish it, until the satellites come within the field of vision. Then wait until the im- mersion or emersion occurs. If immersion is to occur, expect its disappearance a little before an apparent contact. Both immersion and emersion will appear suddenly. Note the instant of its occur- rence, by the waitch. Then calculate the difference in time between its occurence and the time set in the Nautical Almanac. Allow 15 degrees of longitude for every hour, and the same proportion for minutes and seconds, and you have the degrees and minutes of long- itude from Greenwich at London. Sec. 177. In taking the longitude by lunar oiservations, a good sextant, or reflecting quadrant, and a good time-piece, are neces- sary. Look into the Nautical Almanack, and find the angular dis- tance between the moon and the sun or aplanet, or one of the nine fix- ed stars, which are used for this purpose, which may be seen at the time of night or day required. These stars have been selected so as best to accommodate every part of the earth, and to be of suffi- cient magnitude for observation. They are called a (alpha) of Arietes — a (alpha) of Aldeiaran — Pollux — Regulus of first and second magnitude — Spica, first magnitude — Antares — AquilcB re- markably bright — Formalhaut, smaW^Pegasus, a and h. By ex- 94 amining these stars on a celestial globe, or map, particularly Burrit's Atlas of the Heavens, the student may soon make himself sufficiently familiar with their relative positions, to find them at one glance of the eye. In taking lunar observations, half an hour of shewing is better than ten days of reading. The sextant must be set according to the Nautical Almanac, for the nearest third hour. As the time approaches, look at the moon and sweep for a star ; but look at the star and sweep for the moon as they approach each other. Note the instant they touch, according to the almanac and watch. Then calculate the longitude by allowing 15 degrees for every hour's difference between your time and the time given for Green- wich at London. Students who have no experienced teacher near, must read the directions given by Bowdiich. Hours of the day are reckoned from noon to noon ; counting from noon, to 23 o'clock and 59 minutes. Parallax and refraction must be allowed according to the tables at the end of this treatise. Sec 178. Should a surveyor be called to lay offa piece of ground of great extent, (as the Oblong, taken from Connecticut last cen- tury and joined to New-York,) which was to be a true north and south parallelogram, he would be under the necessity of calculating the breadth of a degree of longitude at the north and south ends of the tract, and projecting it upon Mercator's method. To find the breadth of a degree of longitude at any degree of latitude, state thus : As radius at the equator is to 69.1 miles, so is the co-sine of the degree of latitude to the measure of a degree of longitude at the given degree of latitude. Nat. sine of 90®. Miles. Nat. co-sine of 40*. Asl.pOOOO : 69.1 :: .76604 .76604 2764 41460 4146 4837 52.933364 Answer, 52.93 miles. 95 RAIL.ROAD SURVEYING.* Preliminary Survey. Sec. 179. A full party for the field operations of a preliminary survey is composed as follows : One Chief Assistant Engineer, " Compass-man or Surveyor, " Assistant do. " Leveller and Assistant Leveller, " Rod-man, Two Chainmen, " Ax-men, One Flag-man. Sec. 180, With these the chief assistant goes into the field. He is supposed, of course, to have been previously made acquainted, by the chief engineer, with the general direction of the proposed rail- road, and some of the principal intermediate points through which the line is expected to pass. Sec 181. The principal objects of a preliminary survey are, to ascertain the distance between any given points, the difference of elevation between those points, and also the intermediate ground traversed, together with a general sketch and description of the dif- ferent lines pursued in reaching the desired place. Sec. 182. The distance and difference of elevation of any two points are necessary to enable the engineer to determine whether the rise or fall per mile, or the grade, as it is technically called, can be such as will render the motive power proposed effective. The beai'ing of the lines, together with the topographical sketches and field notes, ai'e indispensable in determining the quantity or degree of the curvature. Sec 183. After being made acquainted with the general direc- tion of the line, the chief assistant traverses the ground, examines it carefully for some distance in advance, and having determined upon the ground upon which he will run, directs a flag to some point as far from the place of beginning as it can be distinctly seen, and the * This arlicle was obligingly furnished by Engineer Surgent. 96 assistant at the compass takes the courses as in ordinary land sur- veying, laying off the line in stations df two hundred feet each, by means of stakes driven firmly in the ground and numbered, making No. 1, 200 feet from the point of commencement, and so on. During this process the chief assistant carefully sketches the topography upon each side of his line, directs bearings to be taken to the most pro- minent objects in the vicinity, notes the character of the soil, spring runs, or streams of water that cross the line, and determines the size of a sluice or drain necessary to pass the water. This done, the instrument is moved forward to the point where the flag stood, or some station in line with it, and the course tested by a back sight along the line of stakes to the point of commencement ; when the same process is repeated, unless from the formation of the ground it is necessary to change the course. If this is the case, an angle is made of such magnitude as may be directed by the principal officer in the field ; having due regard to the effect that will be given by tracing a circle between the two lines of which they will be tan- gents. Sec. 184. The level follows the compass, using some point near the commencement as a base, and taking the relative heights of each station. This is usually performed by setting the level at station No. 2, and directing the rod-man to hold his target on the point es- tablished as a base, and to move the vane as directed until the level- ler exclaims,jras!5. The rod-man makes fast and re'plles fast ; when the leveller again looks, and if tiie horizontal hair of the instrument corresponds precisely with the middle of the vane, calls right. The rod-man then carefully reads from the rod the feet and decimals above the surface, and calls the result in a quick, sharp but distinct voice. This the leveller, assistant leveller, and rod-man, enter in their books, under the column of B-sights, and the rod-man moves forward to the next station and holds up on the surface, observing that he gets the natural range of the ground ; when the observation is repeated as before, the result called, and entries made under the head of Fore-sights. Each then takes the difference and places it under the head Above, if the back sight is greatest, and under Be- low, if the fore-sight is greatest. The leveller calls the result, and the assistant and target-man assent or dissent, as their results agree or disagree with his. This done, the last Fore-sight is brought down 97 and placed under the head of Back sight, and an observation taken to the next station, the result of which is placed under the head of Fore sight, and the difference again taken ; and if the back sight is greatest, added to the last result, if less deducted. Again the last fore sight is brought down, and the same process repeated, giving the result for station No. 3 ; and the rod-man goes to No. 4, and directs a small pin to be driven, about six inches from the stake, firmly and close to the ground, upon which he places his rod for the observation, which when taken finishes the business of the "set," and the leveller moves forward, takes his station at No. 6, and re- peats the process before described ; unless, as is frequently the case, the undulations of the earth prevents his getting his sights from the regular stations. When this is the case, he avails himself of the most favorable position, having command of the most stations, and giving equal distances between the 2^sg driven and the one upon which he again proposes to shift his level. The annexed table shows the manner of entering the field notes. Dis- Back Fore Differ. No tance. Sig'it. Sight. ence. Above. Below. Remarks. 1 2U0 8.204 3.30 +4.904 4.90 Start on the sur- 9, « 3.30 4.70 —1.40 3.50 face of the rail at the 3 a 4.70 2.60 -{-2.10 5.60 west end of Troy Bridge. -Peg. 4 ii 2.60 1.406 1-1.194 6.794 5 a 9.465 6.54 + 2.925 9.719 6 a 6.54 4.75 + 1.79 11.509 7 4.75 5.302 —0.552 10.957 Bench No.l on hick- orr tree west of line. Note. It will be observed that the instrument has been changed; but the same process is necessary as in the preceding case, and the only difference is the relative position of the peg to the base first started with. The level from elevated positions is usually turned in various directions, to render it certain, the eye of the engineer has not been deceived in selecting the general route. Sec 185. After the field operations cease for the day, the level- ler will examine his notes, comparing them with his assistant and rod-man, and foot his back sights and fore sights, to see if the differ- ence correspond with that obtained in the field, also add or deduct, 13 98 as the case may require, to or from the starting point, and compare the result with that obtained at the point of leaving off. SeCo 186. When a line for a rail-road has been traced as above described, the second step is to make a rough, or as it is termed, a working profile, and the engineer proceeds to adapt thereto the best grades it will admit of, and from the notes of the chief assistant, le- veller, and compass-man, together with his personal observations, to suggest such changes, modifications, and improvements, as his judgment dictates, for the guidance of his assistant in executing the ^^ Definitive Survey," which is the next step preliminary to breaking ground. Sec. 187. The level in this follows the compass, as in the preli- minary survey, noting every material deviation in the surface over which the line is traced ; also, the level at the surface of water in all streams that are crossed, together with the soundings, and esta- blishing frequent benches or permanent levels, on stumps, trees, or rocks, adjacent to, and convenient for the future adjustment of the line. The stations are now reduced to 100 feet, Euad when it is necessary to take intermediate levels, which is frequently the case, they are entered in the field notes thus ; No. Dis. tance. Back Sight. Fore Sight. Differ- ence. Above. Below. Remarks. 40 41 100 30 40 30 Sec. 188. The duties of the level thus completed in the field, a second, mor€! accurate and finished, profile is made, the grades adapted to it with the utmost care — the streams represented, also the division lines of farms, by a small flag or spear, and the name of the owner of each separate lot or farm, neatly printed between the boundaries, together with other general remarks, such as " op- posite Bethlehem church," " road to the Shakers," " steam saw mill," &c. &c. Also the ratio of the grade per 100 feet and mile. The profile being thus far complete, and the grades satisfactorily arranged, the cuttings and fillings are next to be made out, and placed 99 along a line, drawn parallel to the base line of the profile — ^the cut- tings being placed above, and the fillings below. Sec. 189. The cuttings and fillings are deduced from the levels and grades in the following manner : No. Dis- tance. Surface. Grade. Cutting. Filling. 470 71 72 100 246.57 246.71 260.00 •25U.87 251.71 252.55 7.45 4.30 5.00 The grade being started at the base with the surface, is readily calculated from the rate which has been previously established. When the undulations of the ground are very abrupt, the interme- diates are sometimes deduced in this survey, but not generally until the next Staking Out. Sec. 190. This consists in laying off" the work and defining its boundaries, so as at the same time to procure the necessary notes for a correct measurement of the quantity of earth to be removed or supplied, and direct the contractor how to proceed with the execu- tion of his work. Sec. 191. The slopes necessary to be given in excavations and embankments, are determined from the nature of the soil. In em- bankments however, it is not usual (as we say,) to make ihem less than 1 J to 1 ; that is, with a 1^ base to 1 rise or perpendicular. These, however, vary greatly according to the views of different engineers, as well as from the circumstances above stated. We will suppose, then, that it is determined to lay off the road for a 15 feet width of bed on the top. Going, then, to station No. 10, we find it marked and also entered in our grade book — 4.00 (4 feet below.) The instrument is placed and the rod sent to the nearest bench, which was marked +2.00 — S. 10, (meaning two feet above grade at station No. 10.) The instrument was placed so as to coiTimand a view of as many stations as convenient, and the sight taken, which was 6.00. This was added, on a bit of loose paper, to the 2 feet, 100 making 8, and the letter T placed next the result, understood tripod* or that the cross hairs of the level were 8 feet above grade at sta- tioil. The leveller then moved up his instrument to No. 10, and casting his eye to the right of the line, judged that the ground rose about 1 foot in 12, and gave the target-man the ring of the tape, and directing him to move off at right angles to the line, entered into the following calculations in his head : (Centre 4 feet. — 1 foot rise leaves 3.— 1+3=4-|- and 4-|- +71 = 12.) At 12 feet then, the rod is held up and sight taken, which proves to be 12 feet. The tripod deducted showed the ground to be 4 feet below grade, and that he had not found the medium and was too near the centre, as running in his ihind again the same calculation, he discovered that 4 feet fill would require HH feet distance from the centre ; so the rod was again held up at IS^ feet, and the hair of the instrument cut the target in the centre, which had not been moved, hence gave the proper point of intersection with the surfoce of the proposed line of slope. The calculations previously gone through with in the head, were now made on paper by all the assistants comparing and agree- ing ; it was entered in the field book as shown in sec. 000. The op- posite or left stake was set ofFin like manner, and the leveller and tar- get-man moved on to station No. 11, and the grade ascending j-*/^ of a foot in a hundred, 0.40 was taken from the tripod, which gave 7.40 as the tripod for No. 11. CURVES.* Sec. 192. One of the most difficult parts of the field operations of a locating survey, and that perhaps in which more skill and judgment is necessary, than in any other particular case, is the changing the direction by curving, in a broken or hilly country; when attention must, at the same time, be paid to keeping upon a given level, or maintaining a uniform ascent or descent. Two methods are in common use for tracing curves. One by suc- cessive deflections with the compass or theodolite ; the other by measuring offsets (secants) from tangent lines. The use of the compass is sufficiently explained in sec. 110, 111, and other parts of land surveying. If a theodolite is used, it should be of * This article was prepared by engineer C. B. Evar.s. 101 the best make and graduated with the greatest possible care. If such is the case, a curve may be traced with a great degree of ac- curacy, in the following manner : Place the instrument on the sta- tion at which it is proposed to commence the curve, see that the zero of the nonius precisely corresponds with the zero of the graduated limb of the instrument, turn the mstrument until the vertical hair exactly cuts tlie row of stakes and flag at the opposite end of the line, then tighten the clamp screw to secure the lower limb from moving, let down the needle and note the course of the line. You are supposed to be running on or near the line previously run in the preliminary^ survey, and therefore know about the number of de- grees contained in the angle at which you are about to trace your curve. From this the quantity or degree of the curve is determined, and the curve, in technical parlance, is named from the number of degrees deflected from the course at each station, thus : if at each station the course is changed in the same direction two degrees, we call it a two degree curve ; if the course is changed three degrees, a three degree curve, and so on. Flaving determined what the de- gree of curve shall be, loosen the screw, which connects the move- able and stationary plates of the instrument, and turn the moveable part in the direction you wish to curve until the zero of the nonius corresponds with half the number of degrees on the graduated plate that you have determined upon as the quantity of the curve, thus: If you propose to trace a two degree curve, after setting the instru- ment upon the line as before, you will turn it round until the zero of the nonius cuts 1, or the first division on the card. Then let the chain be straightened from the station at the instrument, and a stake driven in line with the instrument as last set. This will be the first station in the curve. Then move the instrument until zero of nonius cuts 2° on the card, or one degree further than when the last stake was set, and at the end of another chain set another station, and for each station that you set while the instrument remains where you first placed it, you turn it one degree on the card. In this way you may set 8 or 10 stations in the curve without moving up with the instrument. But it must be borne in mind, that the farther you pro- ceed from the instrument the more likely are you to vary from a true curve ; therefore, it is generally advisable not to set more than 8 or 10 stations without moving up the instrument. After setting 102 the stations as above, as far as can be seen with accuracy, move up the instrument and place it precisely over the centre of the last sta- tion set. Turn the instrument back again to zero, direct it to the first station back, and let down the needle to test the course. While the needle is settling, you may calculate what the course ought to be, as follows : Add together the number of degrees deflected in the curve, and add the amount if curving from the nearest magnetic pole, or subtract the amount i? cnw'mgtowards it, thus: The course of a line was N 20° VV, and a curve 2° to the right, was commenced and run with the instrument nine stations, when it was necessary to move up. Sec. 193. The instrument was carried to the last station and set as above directed. While the needle was settling the course was calculated thus : At the first station in the curve the deflection from the tangent was 1 degree, and the 8 following stations were 2 de- grees each, making in all 17 from the course of the tangent. The bearing of the tangent was N 20° W, and the curve towards the north or the nearest magnetic pole ; tlierefore, subtract 17°, the number deflected from 20 the course of the tangent, and it leaves 3° or N 3° W as the bearing of the last chord. On looking at the needle, the course as then indicated was found to correspond with that calculated, and therefore the work was known to be right. Sec. 194. After directing the instrument to the last stake, tighten the clamp screw and turn the instrument 2° on the card for the next station ; but for every station after that deflect but 1°, until the in- strument is again moved up. When the course is . sufficiently changed to stop curving, the instrument must be moved up and placed on the station which you propose to make the end of the curve. After directing the instrument to the last stake as before, deflect 1° on the card, and the theodolite will then be right for run- ning a tangent to the curve. Set the stakes on the tangent as far as you can distinctly see ; but before you move the instrument, test the course and see if the needle corresponds with what you make it by calculation. Sec. 195. Running curves by offsets from tangents, depends upon the same principle as the one above described ; but instead of deflecting with the instrument as above, departure equivalent to the degrees which would be deflected, is measured off from * 103 the tangent or the chord of the last station, produced as the case may be, thus : If you wish to trace a 2° curve by measurement, you will produce the tangent one station farther than the one at which you wish to commence the curve; then, with a tape line graduated to decimals of a foot, measure off the departure for 1° ' straighten the chain from the last stake to the end of the tape, and then set your stake. Then produce the line of the last two stakes a hundred feet farther, and from that point measure off the depar- ture for 2° ; bring the end of the tape to the end of the chain as before, drive your stake, and proceed on in this way to the end of the curve. Compound Curves. Sec. 196. The above methods describe a simple or regular curve, or a section of the circumference of a circle. It is sometimes neces- sary, after proceeding some distance with a curve, to change the degree of it, and curve faster, or not so fast, as the case may be • that IS, a greater number of degrees are deflected at each station^ or the curve made to conform to a shorter radius, thus: if after runnmg some distance on a two degree curve, you find it necessary (to suit the formation of the ground, or from other causes,) to change your curve to 3°, you will proceed as follows : At the station where you wish to change make a deflection of two and a half degrees and at the next three, and so on, as far as you continue the curve' In every case where you change from one degree of curve to ano- ther, add half the difference between the curve you are running and the one you wish to run, to the least, for the deflection at the point of change. Reverse Curves. Sec. 197. It sometimes becomes necessary to change the direc tionofthe curve altogether, and without the intervention of a tan gent, to change immediately from a curve in one direction, to one of a duecfon precisely opposite. When this happens, the operation is as follows: At the point where you wish to change, produce the chord as usual, but do not deflect; drive three stakes in line, and then commence deflecting in the opposite direction. 104 Pencilling and Calculating Cxjkves, Founded upon long Traverses through hilly Districts. Sec. 198. After plotting an extensive traverse, taken for a road, and sketching some of the most important objects minuted in the laeld notes, proceed to pencil out the proposed road, so as to suit the eye or fancy. Then divide off the pencilled road into arcs; each arc extending as far as the curviture continues to be uniform. Then draw a chord line to each arc, and fix the point at each end by mea- suring from the nearest points in the surveyed traverse, if the ends do not fall upon any surveyed angle. As there will, probably, always happen an angle at some point in the arc, find the length of the chord line by the case in trigonometry, where two sides and a contained angle are given. Sec. 199. Consider the angle as moved to the middle of the arc ; for the angle will be the same in any part of the curve, according to a known principle in geometry. Double this angle and subtract that sum from 360 ; and the remainder will be the angle at the centre of the circle of the proposed arc, made by the radii limiting it. Connect these angles by a diagonal line (halving both of said angles,) and this will make four right angled triangles, each with a known horizontal leg ; it being half of the long chord line. Find the length of the radius, as the hypothenuse, in the common manner of proceeding with similar triangles. Sec 200. Having found the radius of the arc, and the length in degrees, (which is the said angle at the centre,) find the measure of the arc in feet thus : Double the radius (making the diameter of the whole circle) and multiply the sum by the formula 3.1416— this gives the measure of the periphery of the whole circle. Then say, as 360° to the whole measure of the periphery ; so the degrees of the arc (as expressed by the angle at the centre) to the measure of the arc in feet. Sec 201. The foot measure, and the degrees of the arc being known, divide the foot measure of the periphery by 100 feet, and the degrees by that quotient. This will give the number of stakes to be set, and the degrees of each isosceles triangle at its apex in the centre, for each portion of the periphery of the arc staked out. 105 Sec. 202. You are now ready to stake out the periphery into hundred feet portions (the usual practice.) Although these are chord lines, unless the curvature is too great for any rail-road or canal, such short chords will coincide so nearly with the curve, that they will come out about equal— or an allowance may be made by shortening the chain a few inches. Sec. 203. The degiees for inflexion (or deflexion from the tan- gent,) at every stake, is to be conducted as follows: Find the direc- tion of the radius, and set the compass on it. Then turn the com- pass around 90°, which brings it upon the tangent line. (The tan- gent hne being always at right angles with radius.) Then deflex from the tangent line equal to half the angle at the centre of the cu-cle, which is made by the radii limiting the 100 fefet chord. But at every following stake, deflex equal to the whole angle at the centre ; from the last line run. Sec 204. Fix the said tangent line as follows : The chord line will form an angle with a line in the traverse ; from which the said chord hne was calculated. This angle can be found in the com- mon way for finding similar angles. Then sight the compass on said traverse line, and turn it through the number of degrees re- quired for bringing the compass upon the chord line. Lastly, turn the compass through the number of degrees found by calculation between the chord line and radius. This brings the compass upon radius, as required. Sec 205. Several other methods are in use among engineers. One is, to find the chord line of half the arc, and the versed sine, by trigonometry, instead of finding the radius as before explained. Then say : As the versed sine is to the said chord line ; so is the said chord line to the diameter of the circle. Then proceed with the formula 3.1416, as before directed. Also halve the diameter to obtain the radius, to be applied as before. Sec. 206. After having the general chord line, the chord line of half the arc, the radius, and angle at the centre, the periphery of the curve may be staked out by measures on the general chord line, and by ofl:sets, as follows : The general chord line and the radii meeting each end of it, constitute a general isosceles triangle ; con- sequently the base angles are known, according to division 4, of section 32. Each staked measure and the radii meeting each end 14 106 of it, constitute an isosceles triangle also, with larger known angles at the base. Subtract a base angle of the former from a base angle of the latter, and the remainder will be one of the acute angles in the right angled triangle, made up of a staked measured line as hy- pothenuse, and the oiFset, and run lines on the general chord line as legs ; which two last lines can be found as in all cases of right angled triangles. After the first stake, the staked measured line will not form the hypothenuse for finding the offsets and run lines. But it must be found by doubling the first angle at the centre, and taking a radius for the middle term in the rule of three. In other respects proceed as before. Continue thus to calculate the run and off*set to the middle of the arc ; and apply the same measures for the other half. Sec. 207. A curve may be staked out by running all the lines from one end, thus : The first staked measure begins at one end, of course. Find the next chord line from the end, as in the run and offset method, before described. All the chord lines may then be found in this proportion : As any one of the chord lines is to the sum of the two adjoining, so is any other chord line to the sum of the two adjoining. Substitute cypher for the outside chord line of the first measured line, for the pui-pose of uniformity, and proceed thus : Suppose P, at the point of beginning. Suppose Q R S T V W, at the terminations of all the chord lines, where the stakes are to be set in the periphery of the arc. Then say : PQ : PO+PR :: PR : PQ+PS 3.90 : 0+7.40 : : 7.40 : 14.04 PS Subtract 3.90 from 14.04=10.14 PR : PS+PQ :: PS : PR+PT , 7.40 : 10.14+3.90=14.04 : : 10.50 : 19.23 2d.( PT Subtract 7.40 from 19.23=11.83 107 3d.^ PS : PT+PR :: PT : PS-f-PV 10.14 ; ; 11.83+'7.40=19.23 :: 11.83: 22.43 PV Subtract 10.14 from 22.43=12.29 PT : PS+PV : : PV : PT+PW 1.83 : 10.14 + 12.29=22.43 : : 12.29 : 23.30 PW Subtract 11.83 from 23.30=11.47 4th. The radius had previously been found to be 6.18. Having calculated all the distances from the station at the end of the curve, the courses only are left to be found. Fix on the tangent Ime as directed in sec. 204. Run the first measure so as to form an angle i.viih the tangent, equal to half an angle at the centre of the circle of the curve, made by two radii limiting said measured line. In this example the angle is , half angle. In running all the other lines, deflect half the said angle (as vi^ith the first measure,) from the last preceding course run. This will bring all the stakes to their true places in the periphery of the curve. Sec. 208. The method of running on the chord line and making offsets, or ordinates, to places for setting the stakes in the curve, de- scribed in the last preceding section, may be calculated from these chord lines, PQ, PR, &c. Call each of these lines the hypothenuse of a right angled triangle, and suppose a vertical leg let fall upon the chord line of the whole arc, and you then have the acute angle at P; of course the run and offset (ordinate) are found. Sec. 209. In running principal, or primary curves, the first me- thod proposed (runnmg on the periphery by inflections towards the centre — perhaps rather deflexions from the tangent,) is in general use. But in staking out the sub-curves, offsets, usually called ordi- nates, are chiefly used. The chord line of a sub-curve is now made, by most practising engineers, just one hundred feet in length. This is subdivided into portions of five feet each. Ordinates are thence set off" to the places for the exact location of the curve. These may be calculated by one of the preceding rules, or by sec. 211 to 215. 108 Sec. 210. The Ordinates for staking out the sub-curves, being very numerous, the calculations are tedious. It is on this account that a 100 feet chord is assumed, as a common length for the chord of a sub-curve, and 5 feet as a common length for the distance be- tween ordinates. This enables the engineer to calculate a table of -ordinates, to fit all cases ; and thereby to save much labor. In addition to this advantage, he avoids perpetual errors, which might be committed by less accurate assistants. Table of Ordinates. This table runs no farther than to the middle offset of the sub- curve, (usually called the versed sine,) because by inverting the order of the ordinates, the other half may be similarly staked out. The calculations for the lengths of the ordinates, are made to every degree and half degree of the first deflection from the tangent, from 1° to 14°. This deflexion is always equal to half the angle at the centre of the circle of the curve, made by the meeting of the two sup- posed radii, limiting a sub-curve. 109 TABLE OF ORDINATES. 05 s Pri. mary An- gle of Def. 6 'V O 6 t-i o Ft. CO 6 O d "P O d o d O d O 00 d O d O D.|M. Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft. 1 14 30 .60 1.12 1.61 2.02 2.37 2.65 2.87 3.03 3.13 3.16 2 14 .58 1.10 1.56 1.96 2.25 2.57 2.48 2.78 2.93 3.00 3.05 3 13 30| .56 1.06 1.50 1.89 2.21 2.68 2.83 2.89 2.94 4 13 .54 1.02 1.45 1.82 2.13 2.38 2.58 2.73 2.78 2.84 5 ~6 12 12 30 .52 .98 1.39 1.75 2.05 2.29 2.48 2.62 2.68 2.72 .50 .94 1.34 1.68 1.97 2.20 2.38 2.52 2.57 2.62 7 11 30 .48 .90 1.28 1.61 1.88 1.80 2.11 2.02 2.28 2.41 2.47 2.51 2.40 8 11 .46 .86 1.22 1.54 2.18 2.31 2.36 9 10 30 .44 .82 1.17 1.47 1.72 1.93 2.08 2.20 2.26 2.29 10 10 .42 .79 1.11 1.40 1.64 1.83 1.98 2.10 2.15 2.19 11 12 9 "9 30 .40 .75 1.06 1.30 1.56 1.74 1.88 1.99 2.05 2.08 .37 .71 1.00 1.26 1.47 1.65 1.78 1.89 1.95 1.97 13 14 Is 8 "8 30 .35 .67 „94 1.20 1.39 1.56 1.69 1.78 1.84 1.86 .33 ..63 .89 1.13 1.31 1.47 1.59 1.68 1.74 1.75 7 30 .31 .59 .83 1.05 1.23 1.38 1.49 1.57 1.63 1.64 16 7 .29 .55 .78 .98 1.15 1.28 1.39 1.47 1.52 1.53 17 18 l9 6 ~6 ~5 30 .27 .51 .72 .91 1.07 1.19 1.29 1.37 1.41 1.42 30 .25 .23 .47 .43 .67 .84 .99 1.10 1.19 1.26 1.30 1.31 .61 .77 .91 1.01 1.09 1.16 1.19 1.20 20 21 5 ~4 30 .21 .40 .56 .70 .82 .74 .92 .99 1.05 1.08 1.09 1 .19 .36 .50 .63 .83 .89 .95 .97 .98 22 4 .17 .32 .44 .56 .66 .73 .79 .84 .86 .87 .77 23 3 30 .15 .28 .39 .49 .57 .64 .69 .74 .76 24 25 3 ~2 .13 .24 .33 .42 .49 .55 .59 .63 .65 .66 30 .11 .20 .28 .35 .41 .46 .50 .52 ..54 .55 j2e 2 .08 .14 .22 .28 .33 .37 .40 .41 .43 .44 .33 T22 2' ^ 1 30 .06 .10 .16 .21 .25 .28 .30 .31 .32 2^ 5 1 .04 .07 .11 .14 .16 .18 .20 .21 .21 110 Sec. 211. To understand the table of ordinates, you should learn the method of calculating it. Proceed as follows : To have a clear view of the subject, plot an isosceles triangle. Make by any scale, tne oase 100 feet according to the practice of civil engineers in this country — this being a chord Hne of an assumed, or given, curve. Make the two equal sides at random ; but at least three or four hundred feet. As the degrees of deflection of this chord, from tan- gent, must be given, [see sec. 206] double this and you have the angle of the apex of the isosceles triangle ; which is the angle form- ed by the two limiting radii of the curve, at the centre of the circle, of which it is an arc. Then by trigonometry say : As the angle at the apex is to the hundred feet base line ; so is half the remain- der, after subtracting the angle at the apex from 180°, to the length of the radius. Sec. 212. Double the radius gives the diameter of the circle of which the curve is an arc. Then add 50 feet (half the given chord line of the curve,) to half the diameter. Subtract that sum from the whole diameter ; then multiply that sum by the remainder, and extract the square root of the product. This will give a standing ordinate, to be subtracted from all future ordinates to be obtained as hereafter directed ; which remainders will be respective ordinates of the above table. Sec. 213. The mathematical principle on which these calcula- tions depend, is this : Wherever the diameter of a circle may be cut, the two parts being multiplied together, and the square root of the product being extracted, produces the ordinate erected on that point where the diameter is cut. Sec. 214. As all the ordinates of the curve will be longer than the ordinate obtained, as in the last section, by the, distance from the given chord of the arc to the centre of the circle ; when it is sub- tracted from the ordinates extending from the centre of the circle to the curve, the remainder will be the length of the offset ordinates in the table. Sec. 215. " The angle at the centre of the circle, of which the rail-road curve is an arc, is double the angle of deflection of its chord line from the tangent." This principal has been so often referred to, and is so important to the engineer, that (contrary to my general plan,) I will here demonstrate it. Ill Make the said chord line the base of an isosceles triangle ; and let the radii, limiting it, be the two equal sides of the isosceles. And let the angle at the meeting of the radii in the centre (say 16°) be subtracted from 180° — leaving 164°. Halve this, making 82° for each of the base angles. Now it is manifest, that if 82° be sub- tracted from 90° (the angle formed by the tangent and the radius,) it will leave 8°, the angle between the tangent and the base of the isosceles triangle ; which is the assumed chord of the curve (or arc) proposed. Sec. 216. Cases may occur, where a cuitc may be required of the form of the long side of an oval. In such a case proceed in all respects as with a circle, until the radius and general chord line are found. Then shorten the radius by calculation, at the middle of the curve, as far as may be required. Consider the remainder of the radius as half the conjugate diameter of an ellipse. Also, the whole radius, doubled, as the transverse diameter. Offsets from the trans- verse diameter (ordinates) may be calculated thus : As the square of the transverse diameter, is to the square of the conjugate; so is the rectangle of the two abscisses of the transverse diameter (sup- posed to be cut where the offset stands,) to the ordinate or set-off. (See farther explanation of the ellipse hereafter.) Sec 217. Rail-road curves must not be too short, on account of the friction of flanges, and danger of running off ; and experience must limit the highest admissable degree of curviture. They are generally compared by length of radius. But in absolute strictness they ought to be compared by a method, in part analagous to the principle on which the movements of planets in their orbits are com- pared — ^that is, similar areas with proportional lengths of arc. The arcs of the areas may be thus compared : Having first found the radius, length of the arc, and area, of the tried railway, double that area, and divide the sum by the radius of the proposed rail- way ; which will give the length of the arc. The proportional lengths of the arcs wUl give the most simple and direct method of comparing them, with a view to their fitness, — ^the longest arc being propor- tionably the most curved and most objectionable. Sec. 218. The Convexity of the Earth is such, that the level- uig instrument, when pointing to a great distance, will cause a line '0 rise above the true level— theit is, the line will be 7.9 inches far- 112 ther from the centre of the earth at the distance of a mile, than is necessary to constitute a level. In truth, by a level we mean a curve, which if continued, will form a circle around the earth, every where equi-distant from its centre. At the distance of two miles, the levelled line will differ from the water level of the earth 2 feet 7.9 inches — at four inlles distance, 10 feet 7.3 inches — at eight miles distance, 42 feet 6.6 inches.- These calculations were made on the supposition, that a tangent line nearly coincides with the 150 thousandth part ofa circle (as 42 feet 6.6 inches amount to about that proportion of the earth's periphery.) Therefore this rule will be sufficient for all cases in practice. Square the semi-diameter of the earth, and the superficial measure of the distance run, sepa- rately — add these squares, and extract the root of the sum. This will give the length ofa line from the centre of the earth to the le- velled line (tangent.) Subtract the semi-diameter of the earth from that obtained line ; and the remainder will be the perpendicular height of the end of said line. Sec 219. If the length ofa degree of latitude (69.1 miles,) be calculated by the square root, according to the preceding rule, it will give 3211 feet 3.5 inches — whereas the true calculation by sines, &c., gives 3192 feet 9.7 inches. The error then in a degree of latitude will be about 18 J feet. Sec. 220. If perfect accuracy is required in great measured lengths on the earth's surface, as 10 degrees of latitude, find the true length of the tangent line, and its elevation above water level, thus : Turn the measured length into degrees, by saying ; as 25000 miles gives 360°, what will 691 miles give ? Answer 10°. Now take this 10° for the angle at the centre of the earth, between the two radii, limiting the measured degree. This gives an isosceles triangle, with 10° at the apex and 85° at each of the other angles. Subtract the 85° from 90°, which gives the angle outside of the isos- celes, between its imaginary chord-line base and the tangent. Then subtract the other 85° from 180°, which gives the angle outside of the isosceles triangle, between said imaginary chord line, and the secant extending the radius line up to the tangent. As the chord line at the base of the isosceles triangle is found by the given pro- portions of it, it follows, that all the angles and one side of the out- side triangle being found, the true length of the tangent and radiu.g 113 extended may be found. Fi-om the extended radius subtract the true radius, which gives the elevated end of the tangent with accu- racy, MEASURING EXCAVATIONS AND EMBANKMENTS. Sec. 221. Under the head of mensuration, the inethod of calcu- lating a parallelepiped, a pyramid, the frustrum of a pyramid, and a triangular prism (including the wedge,) were shewn. Engineer D. C. Lapham, has shewn us [Sil. Jour. v. 27, p. 128,] how to ap- ply Professor Day's eighth problem in mensuration, so as to give a solution in cases, where part or all of these solids are found combin- ed in a prismoid embankment, or in a prismoid mass of earth to be excavated. In truth it is a rule of most extensive generality, apply, ing in all cases where there are straight sides ; or where sides can be equibly averaged so as to approximate plane faces. Wagon- loads of coals with boxes sprung between the stakes, heaps of rough stone, ledges of rocks required to be cut down, basaltic hills, trunks of rivers, &c., may be calculated by it, with more accuracy than by any other hitherto discovered rule. Sec. 222. Rule. Dimde the mas.? to be measured into so many sections, or prismoids, tliut each side shall le nearly straight from one end of each section, to the other. Find the area of the middle of each, and of both ends. Take the middle area four times, and each of the end areas once. Having added these six areas, multiply the sum by one sixth of the length of the section — this gives the solid contents. Sec 223. Without giving a full illustration, it will be a leading thought towards an illustration, to refer to the v/edge and frustrum of the pyramid. If the edge of the wedge is wider or narrower than the head, the width of the edge and of the two corners of the head must be added, to obtain an average of this modification of the triangular prism. Double the area of the triangular wedge is ob- tained by muUiplying the length by the thickness of the head. Thus, having double the area, and three times the length, (calling it a triangular prism,) by multiplying these dimensions together we obtain six times the soHd contents. Therefore it must be divided by six. The frustrum of a pyramid is a parallelepiped, four wedges 15 114 and four pyramids. If the mass to be calculated is a parallelepi- ped, to take its area six times and then divide it by six, will produce the same result, as if but one area was to be used. But if the mass requires a wedge to be sliced off, to reduce it to a parallelepiped, the six areas will include the wedge, and not alter the calculation of the parallelopiped. As all masses with straight sides, &c,, (and which can be reduced to such by judicious averaging,) may thus be calculated ; this seems to be an exceedingly useful rule to the engineer. For it applies equally well to calculating the supply of water per second, &c,, by a running stream — considering the sur- face of the water equivalent to the level bottom of a canal ; and the iottom of the stream as equivalent to the uneven surface of a section of a canal. Sec. 224. In taking the transverse areas of the masses to be measured, the levelling instrument is essential. A plain is assum- ed as the bottom of a canal or as the basis of a rail-road, &c., to be excavated ; and its assumed depth is to be estimated from a fixed chair, (as it is technically called.) This consists of a stake driven strongly into the ground, or some other permanently secured object ; intended as an index of reference, of a known height above the level of the bottom plain of the canal, &c., to be excavated. Stakes are set in the centre of the canal or rail-road ground. Also in transverse sections, where areas are required to be cal- culated. These stakes have marks upon their levelled points, shew- ing their respective elevations and depressions, in relation to the ob- ject of their being set up. One hour's shewing, with the instru- ments in hand, is of more value than many days of reading. The manual use of instruments I shall not attempt to describe. A few general directions will be given in the proper place. Sec 225. In calculations for obtaining cross-areas, we rely upon these two propositions. 1st. Areas of trapezoids are found by add- ing opposite parallel sides, halving the sum, and multiplying the half sum by the distance between them. And, 2d, that a triangle made upon any line, wiil not change its area by moving its apex to any point on an opposite parallel line. [See sec. 33, 1 and 2.] If an excavation for a canal, diverging upwards, is to be made along a side-hill, so that the bottom will be but a foot or two below the sur- face of the earth at the lower side, and eight or ten feet \telow the 115 surface at the upper side, the transverse area may be found thus : 1st. Cut ofFa trapezoid at the bottom, up to the level of the surface of the earth at the lower side ; and cast its area by the first propo- sition above. 2d. Find the level of the surface of the earth at the upper side ; and multiply the upper side of the trapezoid by half the distance between it and the line of the level, which will give the area of all above the trapezoid. Sec. 226. If the earth is undulating or ridgy, these rules may be applied so as to meet every case, with a little exercise of the inven- tive powers. For example : After taking off the trapezoid, as in the last section, if there is a ridge at the surface, take the level of its top and consider it the apex of a triangle whose base is the upper line of the trapezoid. Then if earth is left on one side of the apex, or even on both sides, the upper level may be considered as the base of a triangle, with its apex on the upper line of the trapezoid. And in some cases a trapezoid may extend to the level of the highest ridge, or knoll, and be cast as such. Then the vacant places be cast out or subtracted by making triangles or other regular figures, based upon the highest level. Sec. 227.* The measurement of excavation, embankment and masonry on rail-roads and canals in this State, is usually reduced to cubic yards ; while the latter item in some of the Middle States, is represented by perches of 25 cubic feet each. The method of de- termining on a side hill, the point at which the slope of excavation or embankment would meet the surface of the ground, was explained under the head of Staking out. At the same time, all notes neces- sary for the calculation of cubic yards in excavation and embank- ment, are taken and entered in the field book, as follows, viz. : Stat. Dist. Left. Centre. Right. Dist. 1 Cut. Cut. |Dist. 40 41 100 25 75 14.60 18.40 13.00 +4.6 + 8.40 + 3.00 + 3.50 +6.25 + 2.38 +2.00 + 4.62 + 1.75 12.00 16.62 11.75 Sec 228. When the ground is level, the point at which the slope * The four succeeding sections were furnished by Engineer Evans. 116 will come to the surface, is found by merely adding the cutting of the centre to one half the width of the road, and laying off the dis- tance at I'ight angles from the centre, if in excavation ; but if in em- bankment, add 1^ the centre filling to one half the width of the road, and lay off as befoi*e. This is correct only when the ground is a plain. For if the surface slopes transversely of the line, it is plain that the filling (if filling it is,) will be greater on the one side and less on the other, as you depart from the centre line. And if the filling is greater, the width of the base, or the distance from the cen- tre to the outside of the embankment, will be increased in proper- tion to the slope of the bank — which we have before said, should be l^^ to 1 in ordinary cases. The object then of staking out, under the circumstances above described, is to ascertain the point at which the distance from the centre equals once and a half the filling at that point added to one half the proposed width of the road-bed. Sec. 229. It frequently happens on side hills, that there is filling at the centre stake, but within a few feet the ground rises so much as to require cutting. When this is the case, it is necessary, not only for calculation but also for the convenience of the contractor in commencing work, to ascertain the distance from the centre at which the cut commences. Suppose the instrument set as before described under Staking out, and the elevation by grade at each sta- tion also known. Subtract the elevation by grade of the station in question from the height of the instrument, and set the target to cor- respond with the difference. Let the target-man then hold his rod upon the ground at a short distance from the centre, and move up the hill at right angles to the line, until the hair of the instrument cuts the vane, and the place where the rod then stands will be the point where the cut commences. The distance from this to the centre stake must be measured and noted. Sec. 230. It has been a universal practice on public works, to require contractors to haul the earth a given distance from an ex- cavation, before they receive pay for the same as embankment. This distance, however, is not by any means uniform, but varies greatly on works conducted by different engineers. On all our Slate canals, it is fixed vX 100 feet; while on many of the rail-roads in this State, it is extended to 500 feet ; but we consider a mean between the two to be preferable, and have therefore established it 117 at 300 feet. After calculating the whole amount of excavation and embankment; whatever of the latter item comes within this distance must be deducted. CANALS. Sec. 231. Under rail-roads I have included most of the calcula- tions required for canals. These calculations I shall not repeat. Therefore, the student is to expect but little under this head which appertains to the mathematical arts, excepting what relates to items where water is an agent. This article will, therefore, be chiefly devoted to general descriptions ; excepting that it will close with the necessary calculations on supply of water and filling and empty- ing locks. (See sec. 221—226.) Sec. 232. Navigation. The general term for all transportation or conveyance by water, is navigation ; which is divided into natu- ral and artificial. But as natural navigation scarcely comes within the province of the engineer, I shall take no farther notice of this distinction. Sec. 233. Moving bodies on water. The particles of water move over each other without much friction or with very little adhesion. Therefore heavy bodies move on the surface of water with little resistance. One man has moved 100 tons 7 miles in one day. But as all heavy bodies sink into the water which sustains them, until they displace a measure of water of a weight equal to their own weight ; it is manifest that a volume of water forward of the moving body must be displaced by it. It follows that the form of the body, and the place to which the water is to be removed, are important subjects for the consideration of the engineer. Also, that a boat carrying 50 or 100 tons, will not add one ounce to the pres- sure on an aqueduct bridge, while crossing it. The law which governs ship-builders, in giving form to vessels, is manifest in com- paring the movements of a log or raft in water, with the sharp-built skiff or Indian canoe. Sec 234. Suppose the moving body, for example a crib-boat of lumber, to be 14 feet wide, and the canal the same width. Suppose the crib-boat sinks 3 feet into the water. The water before it is a wall 3 feet high ; all of which must pass back by the stern, when 118 the crib is in motion, through the thin crevices on each side and be- neath. The time required for this escape of the " wall of water" will be such as greatly to impede the progress of the crib. But were the canal 28 feet wide, the wall of water would escape late- rally where there was but little resistance. Still the banks would present some resistance ; as the waters nearest to the boat would be met by the waters stopped by the resistance of the banks. Hence it follows, that great breadth of water is favorable to the movement of heavy bodies on its surface. This principle is tested by our canal boats, when they pass by an artificial basin. Sec, 235. General law for constructing boats or other moving bo- dies. The inclined plane and wedge are known to be mechanical powers, which give an advantage, directly as the length exceeds the breadth. Call the wall of water the resisting force, and horse- power, wind, &c., the power. Then the power will be to the resis- tance, as the greatest breadth of the boat to the distance from the place of its greatest breadth to the extreme fore-end, where it cuts the water. Consequently the longer the boat is, forward of its greatest breadth, the less power will be required to move it. But there are numerous other circumstances to be taken into view in practice, which every ship-builder understands. Such as, that the in- clined plane principle applies to the ascent of the fore-part of the boat. For example, the scow, which retains its full breadth from end to end, has the advantages of the inclined plane in its ascending sled- like fore-end. Mathematicians have said that water would present a solid resistance when compressed with a velocity equalling 18 miles per hour. But this applies when a broad plain is presented ; as if a plank- work should be fixed to the stem-piece of a steam- boat of equal area to a transverse section at the broadest part. Sec. 236. Choosing the ground for a canal. In choosing the ground for a canal, where there is an opportunity to make a choice, the engineer should recommend to the directors the following con- siderations : 1. Course of the prevailing winds. 2. Kind of soil through which excavations are to be made. 3. Its termination in regard to commercial advantages. The illustrious Clinton told me he would not recommend a canal in the present age "which should terminate with its last excavation." Canals should be constructed for connecting navigable waters, or for connecting a navigable wa- 119 ter with a great coal-bed, mine, or quarry. Prevailing winds have not been duly estimated. Side winds, though favorable to all sailing craft, are very unfavorable to canal navigation. They always drive boats ashore, and give no assistance to its progress. Winds in the direction of the canal present an opposing force and an accelerating force, which counterbalance each other on the whole. Northerly and southerly winds being most prevalent in America, east and west canals are not so favorably situated in this respect. It is well un- derstood by all boatmen, that winds are a less impediment (they are always an impediment) on the Champlain than on the Erie canal. Sec. 237. Excavations made in plastic clay, marly clay, marine sand and crag, are found to be permanent. In truth, all stratified detritus, called tertiary formation, make good beds and banks for canals. The marine sand (bagshot sand) would seem to be unfit for canal embankments ; but the trials at Irondequoit and Holley, on Erie canal, prove its fitness. Diluvial and post-diluvial detritus are too variable in character for any general rules, excepting that ultimate diluvian and anallu- vian are good materials for canal beds. Sec. 238. Agriculture and healilu Canals should be constructed with a view to health and agricultural operations. For if they per- mit water to ooze through their embankments, health and agricul- tural operations are injured. Therefore diluvian containing vege- table matter should never be used. Sec 239. Canal hanks, when they are not paved, should be bound by vegetables with creeping roots ; particularly when the canal runs through diluvian, as from Oriskany to near Genesee river. Agropyron repens (quack grass) is probably the best of all American plants for this purpose. It will prevent the production of unhealthy gases, and prevent the banks from shding down in the spring of the year. Banks are secured with well set paving stones where such stones are conveniently obtained. But the quack grass is best in wet places. Sec 240. Stop-dams were formerly made in canals each side of every place liable to failure ; so that when it gave way the naviga- tion would not be interrupted each side of the breach, nor the breach be enlarged by a long continuance of the flowing of water. These dams consisted of planks hinged to a bedded timber in the bottom of 120 the canal, so placed that a strong current would raise them up. Thus a breach in the canal by creating a current would stop itself. They are in some measure discontinued at the present time. Sec. 241. Aqueduct bridges are bridges supporting canals which are carried over vallies, I'ivers, &ev, the canals being, constructed of wood or stone. They are but half the width of the canals, as the boats are never to meet on them. Sec. 242. Culverts differ from aqueduct bridges in preserving the equal breadth of the canal, and in being constructed of earth, like the rest of the canal. Sec 243. Waste-weirs are openings on the sides of the canal, placed at a guaged height, so that the water will waste or flow out of the canal when it would otherwise be so high as to injure some part of the works. (See waste-weir under Water-Power.) Sec 244. Tow-path for men and teams to travel on when towing a vessel or raft which is floated in the canal. It should be three feet higher than the surface of the water of the canal ; and a tow- ing-rope should be 130 feet long. As teams must travel nights and days, and in rainy weather as well as in dry weather, the tow-path should be made of silicious or calcareous earth, that it may remain hard and even at all times. Its bed need not exceed three feet in width ; therefore the cost of a solid bed will be repaid in one season of boating. Sec 245. Cross bridges are made for changing sides with the team, at places where the situation of the canal has inducefl the en- gineer to change sides with the tow-path. These bridges are nar- row ; and the tow-path is so arranged, that the teams cross the bridges without having the tow ropes cast oflf. Sec 246. Heel-path. The side of the canal opposite to the tow- path, is called (by way of a pun upon the word tow — toe) heel-path. The engineer should make the heel-path as good as possible with- out incurring much expense ; for it is often almost as important to boatmen as the tow-path. Whenever a boat comes to for a night, or for a few hours by day, it must be on the heel-path side, and the boatmen will have frequent occasion to use it. Sec. 247. Location of locks is a subject of great importance. In ascending ledges, and in some other situations, the location of locks is fixed as a matter of necessity. But in most cases, the location 121 of a lock is a very important subject for the consideration of the experienced engineer. When the ground is so nearly level that the locks may be at any places within two or three miles, they should be separated as near the usual distances for changing teams as pos- sible. If locks could always be placed at the distance of six or se- ven miles from each other, this arrangement would be the most ad- visable. Long levels should never be sought. Every engineer concerned in laying out the Erie Canal, regrets having laid out the seventy mile and the sixty-two mile levels. By frequently agitat- ing the water of canals by passing it through paddle gates under great pressure, atmospheric air is united with it, and gives it health, ful briskness. The 83 locks of the Erie canal average four miles and a third from each other. The distance between many of them might have been proportioned better. For example, the nine locks at Cohoes Falls* should have been equally distributed to West Troy. This subject is at this day, (1838,) well understood ; and the change is now going on, by averaging the locks along the high ground at the right of the present locks. It was found that the basins between the locks were so much limited in extent, that boats were often grounded by drawing off the water for filling the locks. Sec. 248. Form of locks. The common form of locks is better than the elliptical form. And the size should never greatly exceed one boat in length and breadth, of the largest kinds which are to be used in the canal. For there is nothing gained by passing two boats at once, as the water required for filling is the same. Whereas there will be a great loss of water and time at large locks when but one boat is ready to pass. A still greater objection to large locks is, that larger and longer timbers are required for gates, which are heavy to move and more liable to be out of repair. Sec. 249. River locks. It is generally advisable to construct locks on the s'de of a river, out of the reach of freshets, when they are required for conveying a vessel around a fall or rapid. Thus the sloop lock in Troy (New- York) should have been located. This would have kept sloops out of the current from the fall of water * This article was publislied in 1830. 16 122 over the dam ; and when above the dam they would have been out of the influence of the water-fall, or draft of water as it is called. Sec. 250. A lock consists of side walls, fall walls, hackings, wings, coping stones, recesses with holloio quoins, mitre sills, a pair of gates at each end, and each gate containing paddle-gates and a balance beam. Some locks have a crooked culvert in the wall for each pad- dle-gate ; and some have siphons in lieu of paddle gates. Two or four paddle-gates are most approved ; as one may be opened while the water is low before the gates, and the other after the water has risen to the paddles. Besides a paddle- gate way will weaken the main gate if large enough to let the water through expeditiously ; and it cannot be opened without great power, if large. Sec. 251. Laying out a canal. This may be conducted in all respects Hke laying out a rail -road, as directed under that head ; so far as respects all mathematical calculations and operations. In ge- neral there is not as much care required as to accuracy in turning curves, and minute levelhng by the inch or foot. But the line must be straight to a considerable distance each side of every cross bridge, aqueduct, lock, or whatever lessens the breadth of the ca- nal, by which boats would be liable to strike. Sec. 252. Reservoirs. If there is great necessity for depending on water for the summit level which will be deficient in the dryest seasons, the line should be so run as to afford a convenient place for a reservoir of sufficient capacity. The reservoir should be a very little higher than the canal level ; so that all the water not es- sential for practical use, may be saved in the reservoir to let into the canal from time to time as occasion may require. Sec 253. Water for supplying feeders. In laying out a canal the most important consideration is a sufficient quantity of water for feeding it. To ascertain whether or not the supply will be sufficient, observe ths following directions : How many boats will probably pass every day 1 For a column of water whose base is equal to the square of the area of the lock, and whose height is equal to the difference between the highest and lowest surfaces in the lock, will be required for every boat that passes to the lower level. But we may suppose that only one half of the ascending boats will require the same quantity. Because if a boat ascends first after one has descended, no water is to be taken into account. 123 Sec. 254. Allowance for filtrations and evaporations should be equal to about one-twentieth part of all the water used for feeding the canal. In addition to all the allowances made for the first lock on each side of the summit level, allowance must be made for each succeed- ing lock, which is supplied from the summit level. Perhaps one twentieth for each lock. Two methods are in use for supplying the succeeding locks: 1st. To construct a sluice-way by waste-weir, to carry the excess of water around the upper locks to supply the waste to the lower. 2d. To make every succeeding lock to fall about six inches short of the preceding, in depth. Sec. 255. Supply of water ly feeders. The quantity of water which passes any point in a flowing stream per second, must be as- certained before a decision is made in regard to its sufficiency as a feeder for a canal. A calculation must be made of the supply of water which the stream will afford when it is at its lowest, in summer droughts. The best method of calculation is, to consider the sur- face level of an assumed trunk of the stream, as equivalent to the bottom level of a canal excavation. (See sec. 221.) Then take transverse depths of the stream, and consider them as the upper le- vels for calculating transverse areas of excavations, as taken by the level. (See sec. 222.) Then proceed by the four middle areas, and the two end areas, in all respects as directed in excavations and embankments. (Read sections 221, 222, and 225, attentively.) Sec. 236. The cubical contents of the trunk of water being found as if it was a permanent solid, the time it occupies in passing over its lower limit must be ascertained. If the stream does not exceed about three feet in depth, branching limbs with plenty of leaves may be thrown into the upper end of the trunk, and repeated several times, for determining the velocity. The branches must occupy nearly the whole depth, as water flows faster near the surface. They must be thrown in so as to average the velocity from near the shores to the centre. In deep rivers an empty and filled bottle may be tied togethcB, so that one may float and the other sink. The velocity may be measured by a watch ; but a second pen- dulum 39.1 inches in length, or a half-second pendulum 9| inches is preferable; or 9.77, more accurate. 124 Sec. 257. Having obtained the result of the calculations of the two last sections, proceed to compare the quantity of the supply, with the quantity required for filling the locks the maximum of times required ; allowing the full measure for each filling, and also allowing for all the wastes described in sec. 254. Sec. 258. The time of filling and emptying locks is not connected with the general supply, any farther than to determine whether it will be too long for a reasonable delay of boats. But the paddle- gates for the admission and discharge, is a necessary subject of cal- culation. On referring to sections 139 and 140, and by an applica- tion of common sense, this calculation will be evident. But the dis- charge may require more particular calculation ; therefore a full description of a familiar example will be found in next section. Sec 259. Calculation of the time of emptying the east lock of the pair of locks at the junction of the Erie and Champlain canals. Ft. Length of the lock within. 89 Average breadth within, 15.5 Depth, 14 15.5 217 14 89 620 1953 155 1736 217.0 19313 cubic feet of water contained in the lock. Four paddle gates in the lower gate, each two feet square ; mak- ing each an aperture of four square feet. Reduce each aperture one third, on account of friction (adhesion) and contraction of vein. 3)4.00 1.33 2.67 Reduced aperture. 125 Head is 14 feet ; but falling water diminishes the force of the head one half. Therefore the head is to be computed as if 7 feet high. 7.0000(2.65 2.65 Square root of the reduced head. 4 8 Velocity of one foot head. 46)300 21.20 Length of effused trunk per second, 276 2.67 Reduced aperture. 425)2400 14840 2225 12720 4240 56.6040 Water effused each second from each 4 paddle-gate. 226.4160 Effusion per second from all the gates. 226.4)19313.0(84 Seconds, true answer. 18112 12010 .10056 Sec 260. The calculation made in the last section, was from measures and trials actually made in January 1838, by the engineer class of Rensselaer Institute. By starting all the four paddle-gates at once, and measuring time by the oscillations of a pendulum 39.1 inches in length, they emptied the lock several times, in precisely 84 seconds each time. By often repeating similar measurements and calculations, a very important branch of engineering will be- come familiar. Flumes of flouring mills, factories, &c., are calculat- ed in this manner ; excepting that the square root of the whole head is taken, as in all cases where the water is kept at a uniform head. (See sections 139 and 140.) ROADS IN GENERAL. Sec 261. Kinds of Roads. The principal distinctions among the roads of this country are, 1. Uniedded, most common roads, which receive their forms from the feet of horses and wheels of carriages ; or remain as they were when first laid out. 2. Turn- 126 ■ fiked, when made of earth in the form of beds, descending into late- ral ditches. 3. McAdamized, when made in the same form of turn- piked roads ; but the materials consist wholly of pounded stone, in- cluded between curbs. Sec. 262. Laying out roads is too often under the influence of private interest; otherwise they might always be well laid in new- ly settled countries. But it is very difficult to change the location of a road in an old inhabited town. The ground chosen for com- mon roads, to be supported by tax, should be free from sloughs, and should avoid clay-beds as much as possible. This should be done at the expense of distance ; for the taxes are generally too low for raising large sums for bedding and repairing roads through such places. Sec. 263. Causeys and bridges. So many treatises are before the public on these subjects, that little remains to be said. They should never be at the foot of a steep hill, when it can be avoided by turning the road, or by digging down the hill. And causeys should not be made with large logs or large stones. For the earthy covering will soon be worked down among such coarse materials ; leaving them a naked nuisance. If large stones or large logs are laid in, and covered with small stones or saplins, the evil will thus be remedied. Sec 264. Breadth of causeys and bridges. On the score of economy, as well as of convenience, these should be wider than the custom is, in this country. Bridges will stand much longer for having wide abutments ; and causeys (especially if made of wood) will be more firmly fixed, if wide. Sec 265. Mile and guide-boards. These should never be made of stone. Good sound plank are better. Every one has observed the destruction of mile-stones by mischievous villains. Plank can- not be destroyed, nor even injured, without considerable labor. But a mile-stone is broken and destroyed by one stroke of an ax, or one stroke with a large stone. Guide-boards are always made of wood ; but a mere slip of a thin board, nailed to a post, is soon demolished. A single plank when but two roads meet ; or a trian- gle or quadrangle of planks when more than two, should be used, half-charred at the ends set in the ground. 127 Sec. 266. Planting, or leaving, trees. Trees should be left in laying out roads through woods, and should be planted out in all other cases, in such situations that the road may be shaded from 9 A. M. to 4 P. M. from 1st May to 1st September. The best indi- genous trees m America are the red maple and sugar maple. Trees ^ are of more value than is generally supposed. Take the following calculation : 200 trees may be set at 25 cents each — that is, the 200 trees per mile will cost S50. The cost will be 82000 for 40 miles, a day's travel for a team. On all great thorough-fare roads, 40 loaded teams will pass each day, for 150 days of hot weather. This gives 6000 day -journeys ; which would be but 33 cents each, if the whole expense must be paid the first year. If the public fund only is considered, we may safely say, that less than half a cent is paid each trip for the benefit of dense shades for 40 miles, in the oppres- sive heat of summer. For all such trees will endure 60 to 80 years. Sec. 267. Watering places. The value of watering troughs far exceeds that of shades. Walking horses, which draw loads, should drink once in three miles. Watering places must be supported by wells, excepting in places where there happens to be a stream. To neglect watering places ought to be made a crime by statute, for which road commissioners should be indictable. Sec 268. Level roads. For loaded carriages when the horses walk, a level road is best. But for trotting teams a road is best, when moderately undulating. Even a hilly road, in such cases, is better than a level one. For a load pressing alternately on breast and breech, is easier for the horse. Sec. 269. Resting places. Loaded teams are greatly reheved by resting places on hilly roads. And even pleasure-carriage tra- velling is often benefitted by them. They are transverse mounds of earth, made smooth and sloping ; a little oblique to the direction of the road, that they may serve to direct the water of sudden show- ers into one of the ditches. Sec 270. Zigzag roads. Such roads are required on the face of steep mountains. Such a road takes seven tacks while ascending Catskill mountain to the lakes and ^Mountain House. The principal subject to be considered in la^-ing out such roads is, to give broad spaces for the turns. I laid out that road, under the direction of the 128 commissioners. At first without any regard to the breadth of the turning ground. But I was compelled to alter the whole arrange- ment, after the work was commenced, in the year 1807. Sec. 271. Dug-ways. When roads are cut into side-hills, like shelves, they are called dug-ways. One rule is never to be over- looked in such cases. It is, that the outer side of the road, from the hill, must always be about one foot higher than the inside. And it must not be forgotten, that all springs issuing from the hill, must be carried across, under the road, in very large sewers. This is ne- cessary to prevent an accumulation of ice-ledges across the road, which make it totally impassable. Sec. 272. Agriculture and health. The same rules apply to roads, which are applied to canals in section 238. It may be added that roads are so numerous, as to furnish the means of doing much good in this respect. All the individuals in society make up the whole of community. Therefore if the property of one is benefitted the whole body is benefitted. Numerous cases occur to an honest board of road-commissioners, for doing much good. All conductors of water may be directed so as to benefit the nearest farmer, with- out any injury to the public. So in passing fai'm houses, there are numerous ways for an accommodation. Stagnant waters should never be allowed to settle down from a road near a dwelHng house. Sec. 273. Hills and mountains. The ascents and descents of hills, are important subjects for road-commissioners. By attending to the laws of the inclined plane, and to the balancing principle appli- ed by the horse in drawing a load, common sense will suggest rules of practice. When a horse is drawing to the extent of his strength, his hind feet form a pivot upon which the weight of his body is ba- lanced against the resistance of the load. Should the hill be so steep, that the centre of the gravity of the body of the horse, is di- rectly above his hind feet, he can draw nothing. Reduce the steep- ness of the hill, and the weight of the horse will apply in an increas- ing ratio. This ratio will have the advantage of the lever also. The ascent of 18 inches to the rod is the limit imposed by the legis- lature of the State of New-York, on several turnpike companies in mountainous districts — that is, an ascent of 5 degrees. If the ascent exceeds 6 degrees, it is a tiresome road ; and commissioners 129 ought to avoid any farther increase in the ascent, by the zigzag form. (See sec. 270.) Sec. 274. Angle of friction in the movement of carriages on dif. ferent roads. The angle of friction is estimated by placing a car- riage on an inclined plane in the road to be tested, where the de- scent is just sufficient to give the smallest degree of motion. The motion must not be sufficient to acquire any increased velocity by its progress. A section of this inclined plane is to be considered aS radius and the height of the elevated end as the sine of the angle of ascent. One ton for a load on a M'Adam road, will move with the radius 50 to a sine of one by measure — on a very smooth pavement the radius of 68 to a sine of one. On a level road of the same qua- lity, the same proportional of weight, suspended over a pully, as the sine to the radius, will start the load. Thus about 40 pounds sus- pended over a pully will start a ton load on a level, if the carriage is of the best construction in regard to friction. Sec. 275. Location of bridges. Two considerations must always govern in the location of bridges, if any choice can be exercised. 1. It should be located below a natural ice-break if possible ; as a fall of water, &c. 2. It should be placed where the abutments may be secure ; for where abutments can spread, the bridge, if arched, is never secure. Sec 276. String. pieces. When a bridge is made by planking upon straight string-pieces, they must be strongest in the middle. To be largest is not always to be strongest. If the grains of the timber are straight, and the top of the string-piece straight on the top, it will be but little stronger for swelling underside so as to be much thicker in the middle. For its increase in strength depends on the lateral adhesion of the fibres, which is feeble in the straight- est and most thrifty growing timber. But if the timbers swell out at their sides, this objection will not apply in so great a degree. Sec. 277. When string-pieces are supported on bents or piers, some calculation is required in setting off the distances between the supports, or in selecting timbers of the most suitable length. By referring the mind to stations taken by men, when carrying long and heavy timber, common sense will have a sufficient guide in this matter. For example : 6 men are to carry a heavy beam, which is 60 feet long. To place these men in pairs, so that each pair may 17 130 support equal portions of the weight, their carrying-sticks must be so placed that each pair shall bear 20 feet. If the timber was cut into three 20 feet pieces, and each stick was put under the centre of each 20 feet, it is manifest, that the weight of the whole 60 feet would be equally distributed. To arrange the three pieces in one line, touching end to end, would not alter this proportion — ^neither would pinning, or otherwise uniting them. Therefore the first pair would be placed 10 feet from the fore end — the second 30 feet — the third 50 feet. If two of the men were placed at the fore end, and the other four were to lift at one stick, that must be placed 45 feet back. This would leave 15 feet back to balance the 15 feet be- tween it and the centre. The pair at the fore end, would, upon the lever principle, lift but half the other 30 feet, if the back half was cut off and the other stick placed at the new cut end. But after all the back half is neutralised by the balance of the part back of the hind carrying-stick, the two sticks are applied to the fore half with different levers by 15 feet. That is, the centre of the weight of what is not balanced, is 15 feet from the forward carrying-stick, and 30 feet from the back carrying-stick. Therefore the forward stick will support two thirds (20 feet) and back stick one third (10 feet) which, added to the back 30 feet, gives the true proportions. I preferred this method of illustrating the whole doctrine of beam pressure to that of givmg a set of rules. Not only bridges, but flumes, mill-dams, locks, house-beams, &c., &c., require an atten- tion to this subject. Sec. 278. Repairing roads. It is astonishing that our highway masters and turnpike companies, still continue to fill up ruts with loose soil ; when they see the first wheel that follows such repairs restore the ruts. It is still more absurd to fill ruts with stones, wood, &c. But one efficient method has hitherto been adopted. It is to fill the ruts with the same material of which the road consists, by pounding it down in a succession of thin layers, until it is consider- ably harder than the rest of the road. This makes a durable repair and the labor is not great. Two men with sledge-hammers, one on each side, will pound the fourth of a mile of bad ruts in one day ; while one man with a cart and one with a spade will supply the filling. 131 WATERWORKS. Sec. 279. The term Waterworks is applied to the conduction of water through pipes or raceways, (mostly through pipes,) where water is to be used as an element ; not for its mechanical force. The principles of waterworks, as a science, are not generally stu- died, and, of course, are little understood. Sec. 280. We have but one American treatise ; and that has, probably, no equal on either continent, as a concise digest of all that - is valuable on the subject. I mean E. S. Storroio^s Treatise on Waterworks. To those mathematicians who wish to go deeply into the subject, and to trace this truly experimental science back to its origin, then to follow down its history to the present day, this little duodecimo of 242 pages, is most emphatically recommended. He does justice to the early investigations (in 1771) of Abbe Bossus — of Chezy (1775)— of Dubuat (1786)— of Coulomb (1800)— of M. de Prony (1804.) But the German, Eytelwcin, may be said to have given the last finish to the formulae now in use, in 1814 and 1815, through the Memoirs of the Academy of Berlin. Sec. 281. Waterworks being a subject not commonly studied by American engineers, I shall here give a few essential formulae, with a very general statement of the theory. Sec 282. Water moves in pipes or raceways, under the govern- ment of ACCELERATING FORCES and RETARDING FORCES. In Strict- ness, there is but one accelerating force ; which is the head waters above the place of discharge. And there is but one retarding force ; which is the adhesion against the sides of the conducting PIPE, or raceway. But in calculating practical results, it appears to be necessary to take into view the area of the pipe, or raceway, the interior surface of adhesion, and the length of the raceway, or pipe. Sec 283. It has already been shewn [see sec. 139 and 140] that the force given by the head waters, is as the square root of the height. But the retarding power of adhesion (sometimes called fric- tion) depends, for the estimate of its influence, solely on trial. As different calibres of pipes, and different measures of sides and bot- toms of raceways, present different proportionals of surface for ad- 132 hesion, nothing but very extensive series of experiments could fur- nish rules, or formulae, for practical use. Sec. 284. Though the fall of water (that is, the elevation of the head above the place of discharge,) is elementarily the only accele- rating force, and adhesion to inner surface of the conducting pipe or channel, is the only retarding force ; yet it is found that several mo- difications and combinations of these elementary principles, must be taken into all calculations — for the reasons read Storrow. Sec. 285. In pipes of Avood, iron, earthen, or whatever close con- ductors may be used, the following combinations and modifications are necessary : 1. Diameter of the pipe. 2. Difference of level between the head and discharge of the water. 3. Length of the pipe — consequently the continuance of adhe- sion. 4. Difference of level divided by the length of the pipe. 5. Velocity per second. Here each section is considered as the acquired velocity, or the successive retarding results ; and may be estimated in infinitely small divisions. 6. The quantity discharged per second in cubic feet. The area of a section of the pipe, as divided by the inner peri- meter, requires some calculations. This quotient is called the viean radius. It may be perceived, that the larger the area the greater will be the velocity, but the greater the perimeter the less the velo- city. In truth, after the perimeter is increased to a certain propor- tion, compared with the area, it totally overcomes the area ; and the water stops flowing, by the principle called capillary attraction. Sec. 286. Formula for fipes when quantity discharged is sought. The engineer is called on to answer this enquiry : How much water will be discharged per second, if the head (above the place of dis- charge) is 70 feet, the diameter of the pipe 10 inches, and the length of the pipe 900 feet? Prepare for the rule by reducing the feet to inches. Rule. 1st. Multiply the diameter by 57, and that product by the head (70 feet) and set this last product down for a dividend. 2d. Take the first product (diameter multiplied by 57) and add to 133 it the length of the pipe (900 feet) and set this sum down for a divi- sor. 3d. After the division, extract the square root of the quotient. 4th. Multiply that root by 23.33— the product will be the velo- city in inches per second. 5th. Find the area of the pipe at the place of discharge, and multiply that by the velocity in inches per second. This product is the quantity of water discharged per second in cubic inches. Sec. 287.* Formula for pipes lohen the diameter is sougJit. What diameter of pipe will be required to discharge 3 cubic feet of water per second, if the head is 10 feet and the length of the pipe is 4000 feet? Prepare for the rule by reducing all the measures to feet and decimals of feet. Rule. 1st. Square the cubic feet (3) multiply the square by the length of the pipe (4000 feet) and take this product for a dividend. 2d. Multiply the head by the square of 38.116, (10 x 38.116 x 38.116=14528.29) and take this product for a divisor. 3d. After the operation of division, extract the root of the quotient to the fifth power, by sec. 30 ; which will give the diameter of the pipe in feet and decimals of feet (1.199.) Sec. 288. Formula for open canals when the velocity and quantity are required. How much water will be delivered per second, if the area is 4.8 feet (2.4x2) head 10 feet, perimeter inside 6.8, length 30 feet? Prepare for the rule by reducing all the measures to feet and decimals of feet. Rule. 1st. Multiply the area and head together for a dividend. 2d. Multiply the perimeter and length together for a divisor. 3d. After the operation of division, multiply the quotient by 9582, and to this add 0.0111. 4th. Extract the square root of the last sum. 5th. To the root add 0.109. This will give the velocity of feet per second. Multiply the said feet by the transverse area of the trunk of flowing water, which will give the quantity in cubic feet. Sec. 289. As open canals present their flowing waters to the eye, their laws of motion are subject to more convenient inspection than those of pipes. At the Rensselaer Institute the students in civil en- * This, and three other examples, were obUgingly calculaled for my pupils by my learned friend, Mr. Slorrow. 134 gineering have generally obtained the following proportional results, or nearly so. A raceway, smoothly planed within, with exact slid- ing gates 36 feet apart, is used. It is 2 inches wide and 2^ deep within. When the raceway is so inclined that the upper gate is 56 inches higher than the lower one, the water flows from gate to gate (36 feet) in 6 seconds. If the upper gate is drawn 2 inches, the velocity so far contracts the flowing trunk of water, that the lower gate precisely touches the surface when it is drawn 1.1 inch. There- fore the increased velocity, under an angle of 7° 23' inclination, diminished the volume of water nine twentieths in flowing 36 feet. Sec. 290. These experiments may not be perfectly accurate. But they approximate truth near enough for general illustration ; and students of all schools should repeat them. The law of falling bodies, as illustrated in sec. 140, and the law of the inclined plane, as in sec. 152, should be referred to in explanation of this experi- ment. Students will not overlook the difference between pipes and open raceways, caused by the water on the upper side, in the latter case, not being subject to adhesion. Of course the perimeter in- cludes the bottom and two sides only ; whereas pipes present ad- hering surfaces on all sides, with additional resistance from adhesion on account of increased pressure against the whole inner surface. Sec. 291. As almosplieric pressure often has more or less influ- ence upon the flowing of water in pipes and open raceways, the stu- dent is referred to sections 141 — 144, where the most important principles are explained. The pressure averaging about a ton weight to a square foot near tide-water level, it often becomes a sub- ject deserving particular attention. But on high mountains the pressure of the atmosphere is greatly diminished. It even becomes so rare at the height of about 45 miles, that it does not reflect the sun's rays sufficiently to become visible in the state of twilight. Sec 292. The highest point where the atmosphere is sufficiently dense to reflect light, may be found as follows : 1st. Take the time, by a good watch, between the disappearance of the sun in the western horizon, and the disappearance of twilighto 2d. Calculate this ti7ne, in the manner hereafter explained, so as to ascertain what the time would be (or is) where the sun goes down vertically (as at the equator on the 22d of March.) 135 3. Then say, as the minutes of 24 hours to 360 degrees, so are the minutes, vertically taken, between sun-set and twiUght-set (about 70) to an angle at the centre of the earth, formed by a radius to said centre from the observer, and from the place on the earth where the sun disappears, where the observer sees the last depart- ing ray of twilight in his horizon. 4th. Take half said angle at the centre of the earth, for one of the acute angles of a right angled triangle, formed of the observer's horizon, his vertical semi-diameter of the earth, and a line from the centre of the earth to the point of the last appearance of twilight. 5th. Then say, as the co-sine of the half angle at the centre of the earth, is to the semi-diameter of the earth (about 4000 miles,) so is radius (the angle at the observer) to the distance from the centre of the earth to the point of the last appearance of twilight. 6th. Subtract from the last answer the semi-diameter of the earth (4000 miles) and the remainder will be the height of the atmos- phere, where it is just dense enough to reflect and refract the sun's rays sufficiently for rendering it visible to the earth's inhabitants. Sec. 293. It may be desirable to the correct student, to understand a practical method for determining the true time to be assumed for the calculations of the last section, between the disappearance of the sun's face, and the disappearance of its last departing rays of twi- light. The plainest practical method, and the one best adapted to student's practice, is as follows : 1st. Set a compass to find the bearing of the point of the sun's setting, and note the time of its setting. 2d. In the same manner, note the time and bearing of the obtuse apex of the last departing rays of twilight. Thus you have the dif- ference of time between the disappearance of the sun's face and of twilight. 3d. Call the difference of time between the setting of the sun and the setting of twilight, the hypothenuse of a right angled triangle. Call the horizontal angle between the point where the sun sets and where the day-light sets, the horizontal leg. For the vertical leg, (the answer required,) apply the rules for right angled triangles. Also remember to turn time into angles, as in all cases where 24 hours give 360 degrees, &c. 136 Sec. 294. The said vertical leg may be found without taking the bearing of the points at setting of the sun and twilight, by finding the angle which the sun makes with the horizon at setting. This may be done by calculation, made from the latitude of the place of obser- vation and the sun's decHnation. Sec. 295. As aqueous vapor diminishes the specific gravity of the atmosphere, it often becomes a subject of consideration — particularly in the use of the barometer, and in calculating for the ascent of water in pipes in passing over hills, &c. Vapor being visible in the form of clouds or fogs, it may be well for the student to give his attention to the natural history and heights of clouds, for part of each day during one week ; as clouds are lighter than air. Sec. 296. Five forms of clouds often precede each other in regular series. In fair weather during summer months, the stratose clouds, usually called fogs, often appear in the morning near the earth. After the sun shines upon them, they ascend in a state scarcely visible, and at length form the cumulose clouds. These are the bright shining clouds in brilliant heaps above, with apparently straight bases below, when viewed horizontally. They ascend still higher later in the day, and form the cirrose clouds. These have a fibrous flax-like appearance, and rise the highest of all clouds. At length they descend more or less, and become the cirro -cumulose clouds, by assuming a knotted or curdled appearance at first, and then becoming confluent. They either become stationary, produc- ing rain or snow; or break up, and their fragments become cirro-stratose clouds. These are the patches which have a stratified appearance when viewed horizontally ; but they never approach the earth, like fog. No rain falls from this series of clouds, except- ing while in the cirro -cumulose form. Sec. 297. Three forms of clouds seem to he independent of all other forms, and of each other. The nimhose cloud, generally call- ed the thunder cloud, as soon as it commences forming, begins to move pretty uniformly and steadily. At first it exhibits a heaped top, like the cumulose cloud ; but as it advances in size, it shoots forth a kind of spray-like form from its uppermost heads. It usu- ally produces rain, and breaks up soon after. The villose is a kind of open fleecy cloud, called scud, which moves with great rapidity, often in a direction different from the clouds above. It is generally 137 formed suddenly, and breaks up suddenly. The only remaining variety is the cumuh-stratose clojud. It is very rarely formed, and always appears to rise up in the horizon like the smoke from a fur- nace. Its top generally seems to pass into a cirro-stratose cloud above, and there spreads out like the top of a mushroom ; it is there- fore generally called the mushroom cloud. All snow storms and settled rains proceed from the cirro-cumu- lose, and all hail storms and showers, from the nimbose, clouds. Sec. 298. The height of a nwibose cloud may be taken as shewn in the following example : May 30th, 1837, during a severe thunder shower, I suspended a pendulum near the west door of the Institute, and directed an assistant to watch its vibrations, while I observed the origin of three successive chains of lightning. The assistant noted, by the pendulum, the seconds between the flashes and the sound. The time averaged 21^ seconds — and the angle above my horizontal level was found, by the sextant, to be 11^°. Allowing 1124 feet per second for the sound, the hypothenuse from that point in the cloud was 24166 feet. This gave the height of the cloud 4818 feet above my level. The earth's convexity (after finding the horizontal leg) gave 13.3 feet. (See sec. 218.) Therefore the height of the cloud was 4831.3. But my level was 73 feet above the tide-water of the Hudson — of course the cloud was 4904.3 feet above tide- water level. WATER-POWER, APPLIED TO DRIVING MACHINERY. Sec 299. The elementary laws of water-power are explained and illustrated in sections 136 — 140. The student must attentively review those five sections, when he is about to be exercised in the present application of this power. I shall go no farther into the subject, than is necessary for preparing the student for those duties which strictly belong to the out-of-doors engineer. I mean, that he must be qualified to take the original measurements, and calculate the power of any proposed mill-seat, before the commencement of any of the works. In doing this, he takes flouring mills as his standard ; estimating their powers by the quantity to be floured in a 18 138 given time. Then it is the business of the miU-wright and machinist* to make comparisons, and construct the works according to circum- stances. Sec. 300. The first step to be taken is, to take the necessary measures, and to calculate in cubic feet, pounds, or tons, the water which flows by any point in the stream per second. Directions for this operation are given under the head of Measurements of Exca- vations and Embankments, sec. 221 and 222. But it may be well to give more particular directions here. Sec. 301. To find the supply of water, select a time for taking measure when the stream is at its lowest, highest, or middle state, according to the object of the owner. Sometimes a mere flood-mill is desired — in other cases it is not desired as a drought-mill, &c. Select a trunk of the stream which is the most uniform in width, depth, and velocity, and traverse one shore with the compass. Its length ought to be such, that sticks, leaves, &c., will require at least 10 seconds to flow through it — 20 seconds will be better, if such a trunk can be found, that is nearly uniform in width, depth and velocity. The length being taken in feet proceed as follows. Sec. 302. Take measures for a transverse area at every mate- rial variation in depth, width, or direction, in this manner : Measure the breadth, and also the depth, at every material difference in depth — be particular to notice the distances between the places where depths are taken. Also measure the distances between the mea- sured areas — all in feet and decimals of feet. Sec 303. Take the velocity of the stream in this manner : Sus- pend a pendulum for beating seconds 39.1 inches in length — or use a watch with a second hand. The pendulum is preferable. Let a careful assistant note the seconds. If the water is shallow, throw in branching weeds, green bushes, &c., of such forms that they will be driven along by the action of the stream from the top to near the bottom."!" Note the seconds they occupy in running through, about eight or ten times, in the strength of the current. Then try the same experiment as many times towards each shore ; and use your judgment in estimating the proportional part of water in the trunk * Oliver Evans, and his editors since his decease, have prepared a work, under the title of Mill- Wright's Guide, which surpasses all commendation. It is a remarkable specimen oftlie union of science and art. t Shavings of white wax arc best. 139 where the side experiments were tried — always bearing in mind that the diminution of velocity as well as of the depth, are to be taken into view ; for though the diminution of depth will come into the calculation of areas, its retardation by adhesion at the shallow bot- tom and shore, must be separately estimated, as near as may be. Note. These being all the measures to be taken in the field, you will return and make the calculations. Sec. 304. A plot of the following kind, will greatly facilitate the operation. Fix on the scale by which your plot shall be made ; then draw two horizontal lines at a distance fx'om each other equal to the length of the trunk. Plot the traversed shore of the trunk, which will terminate in the parallel lines. Then lay off all the trans- verse sections, parallel to the horizontal lines, and the true distance from each other. Consider these lines as those drawn across the straight surface of the stream. Let fall perpendiculars from each, according to depth of the measures taken. Coimect the lower ends of these measures, which will exhibit each transverse area. These areas may then be calculated by Lapham's method, as described in sec. 225, or they may be cut up into triangles and trapezoids, as described under land surveying, sec. 94 and 95. Sec. 305. The cubic contents of each section of the trunk may then be cast, by adding the areas of the ends to four times the area of the middle, and multiplying that sum by one sixth of the length. (See sec. 222.) Sec. 306. Having found the cubic contents of the trunk in cubic feet, average the number of seconds, which the branches, &c., oc- cupy in passing through the length of the trunk. Divide the con- tents by the seconds, and the quotient will be the cubic feet which pass by the lower point in the trunk per second. Cubic feet may be reduced to pounds by multiplying by 60, and pounds into tons by dividmg by 2000. Thus you will have the cubic feet, the pounds, and the tons, which the stream supplies every second. Note. Here, as in other cases, I adopt the ton of the revised laws of the State of New. York. Students have only to apply com- mon sense, when they have occasion to adopt the gross ton (2240) and, consequently, corresponding numbers in other weights. Sec. 307. It frequently happens, that a dam is built, and works are already in operation ; but the whole of the water is not employ-. 140 ed. Additional works are to be added ; and the engineer is called upon to estimate the quantity of waste-water, which pitches over the dam. The word Weir, or Waste-weir, is applied to this water- pitch, because such a pitch is in use for passing off the excess of water, in freshets, from canals, &c. It takes its name from the strong wires (weirs in German) which are inserted, to prevent inju- ries, which might occur by drifting over small articles of value. Directions for calculating the quantity of water in cubic feet, which crosses the weir per minute, are given in the next section. Sec. 308. Take the depth of the sheet of water by setting a very thin scale with a sharp edge against the stream, just touching the extreme edge of the waste-board. This measure will be sufficient if the said edge is perfectly horizontal throughout the width of the sheet of water. But the sheet must be divided into sections, and they measured separately, for each change in its level. Take the measure of the whole breadth of the dam, as well as of each sec- tion which you may think it necesary to make. Take the cubic feet per minute, set in the table against the inch of depth. This would be the true answer required, if the sheet of water was but one inch wide and confined by side-boards or walls. But if the sheet of water is 50 feet wide, or of any other width more than an inch, multiply the said cubic feet which pass down in the inch sheet per minute, by the whole width of the sheet, taken in inches. This would give the required answer, were the flow of water the same in a confined situation, as when flowing freely over a broad space. To compensate for this difference, divide the above product by 20, and add the quotient ; which will give the true answer. Sec. 309. If the measure of the depth is found in inches and quarters of inches, take the whole in quarters. As for 2 inches and 3 quarters, look for the cubic feet against 11 inches (as this is the number of quarter inches) and take the eighth part of the said cubic feet, set against 11 inches. Sec. 310. If the table of depths is not sufficient for the measure of the depth of the sheet, take such one of the depths as will pro- duce the measure by doubling, tripling, quadi'upling, &c., and mul- tiply the cubic feet set against the taken depth, by the formula set against said doubling, tripling, *— >. y-^ ^— ^ continued, will return into itself, and form ^■^ ^"^"^ ^ a circle ; as each of the curves in the figure. Circle, is a figure bounded by a continued line, every where equi-distant from a point in the cen- tre, c sector, bounded by two radii and an arc ; d segment, bounded by a chord line and an arc ; a the centre. Ellipse, made by moving one pin's point around two focal ones ; and is kept in the periphery by sliding around in the loop of a thread. Square, is a figure of four equal sides meeting at right angles. Rectangle, (or parallelogram) a long square with opposite sides only equal. The line con- necting opposite corners is a diagonal. Rliomh, a figure with four opposite equal \ sides, not meeting at right angles. Superficies, having length and breadth without thickness. Solid, having length, breadth, and thickness. 23 178 Right angle, a BqaaTecotneT. Sec. 31. Obtuse angle, opens -widet than a right angle. Sec. V 31. jf Acute angle, does not open as wide as a square. Sec. 51. A right-angled triangle contains one right angle. Sec. 32. Ohtuse-angled triangle has one obtuse an- gle. Sec. 32. Acute-angled triangle has all three angles acute. If it has two equal sides, it should be called also Isos- celes triangle. If the three sides are equal, it is also called Equilateral. Sec. 32. a c is the sine of the angle e. Sec. 33, article 3. %/\ V The figured line is a line of chords. Sec. 33, "^rV y article 4. Illustration of the principle that a tri- angle contains 180 degrees. Semi-circle l|~~ g i is the measure of 180° — angle a in it equals angle a in the triangle — e equals e — c equals c. Sec. 33, article 5. 17» Geometrical trigonometry, is illustrated by plotting this figure ; having the side given, whicli is drawn from the angle at 80 to the angle at 70, and drawing the lines up to g, through the marked points on the arcs d and /, until they cross at g. See sec. 34. A / B \ Frustrum of a cone or pyramid, found by calculating the pyramid as if topped out ; and then calculating and subtracting the added point. See sec. 53, article 6. This figure also illustrates the 7th arti. cle, Guaging, under the 53d section. Field surve3ring with a cross, where a farmer is desirous to know the contents of a field for planting seed, to pay for mowing by the acre, &c. See sec. 60 and 61. ISO Map of a farm referred to from sections 74, 75, 78, and then in all the sections to sec. 89. And again from section 106 to 109» where heights and distances and division of land are explained. ■^ 181 Plot of a survey referred to from sections 91 to 95 ; wherein the method of reducing and raising the scale, plotting, triangular cuttings and castings are explained. 182 Plot of a survey, wherein the area is cast up by reducing the whole'survey of a single triangle. This is referred to and explained from sections 96 to 99. Contraction of the vein, from sec. 315. One third of the area is generally deducted for the contraction of the vein. This expressed decimally is 0.666 — this agrees nearly with Bossut's experiments. Eytelwein adopts 0.640. It has been demonstrated by experiment, that a conical tube, whose length is about 0.88, the diameter of its base, if adjusted to the aperture, will reduce the contraction of the vein to about one sixth — that is, the 183 area of the contracted vein will be but one sixth less than that of the area of the aperture. Also, that a gate-hole through a four-inch plank, cut a little convergingly, will add much to the efflux of a co- lumn of water ; by lessening the contraction of the vein. Plot of a survey calculated by the trapezoidal metTioar-Tererred to from sections 100 to 104. 184 Harbor survey ; being a section of Hudson river, above Troy Referred to from sections 120 to 126. ■■■■-'■■ m 'r'W ■■ ■■ Mi -■/.mA 1 -m ||\ 9o:m ■■■' .•■■"■■•■ # /■ ,^3* ■■II \ ■Mi.. ■■■mm ..-•■■ ..-ill " ■■il .ll ■■■/■■¥ ::t6S 185 For sec. 198 to 200, on Rail-Road Curves. Two propositions, referred to in sec. 199, are here given at full lenglli. 1. Two chord lines meeting in the periphery of a circle, form an angle, which is measured by an arc, half as long as the arc requir- ed to connect the ends of two radii, which meet the ends of the chord Imes. As the arc A H C is twice as long as an arc required to measure the angle ADC. 2. Two sub-chord lines meeting at any point in the periphery of a circle, the point of meeting will continue to form the same angle, if moved to any other point of the periphery, on the same side of the general chord line. As the angle 130° at B, is 130° at D. Description of tlie figure referred to sec, 198 to 200. Scale 200 feet per inch. Radius E C 300 feet. General chord hne A C 465 feet. Sub-chord line A B is a traverse line 360 feet long, taken in the field. Sub-chord line B C is a traverse line 132 ^^^^ 2325tf ■■% 4 ■. ^r5tf^• H^'" IOO°E ^c of 260 24 186 feet long, taken in the field. Angle B at the meeting of the tra- verse lines, is 130°; as found by considering the compass directions of each. Moved to D, it continues to be 130°, according to the se- cond proposition above. This angle doubled makes 260°, and is measured by the arc A H C, according to the first proposition above. This subtracted from 360°, gives the angle 100° at E. The diag- onal line E D halves the angles at E and D. The general chord line A C, being halved by said diagonal line, give horizontal legs to four right-angled triangles; to wit, K L M N, each being 232|^ feet. Ahwe description extended to the second figure ; which is referred to sec. 201 to 204, Tan. the tangent fine, from which the line a J is deflexed 10°, a number equal to half the angle (20°) of the isosceles triangle at the centre of the circle. The next line c d is deflexed from the chord line a c 20°, a number equal to the angle at the apex of the isos- celes triangle at the centre of the circle. All the remaining de- flexions are 20° also ; as they are deflexed from the last preceding chord line, which is double the deflexion from the tangent line. IQO' 187 Rail-Road Curve. Illustration of sec. 207. The dotted lines are used in sec. 208. This example comprises more than half a circle ; of course the line P W is shorter tlian P V. In practice no curve ever includes half a semi-circle. Sliding Rule. The sliding rule has four Hnes — two stationary on the wooden part, two sliding ones on the brass slip. The upper one on the wood is marked A — the upper one on the brass is marked B — the lower one on the brass is marked C — the lower one on the wood is mark- ed D, and called girt line. Measuring boards. 1. Take the width of the board in inches. 2. Find the number agreeing with the number of inches on line A, and shde figure 12 to it on B. 3. Read off the square feet by measuring the length of the board in feet and inches, and finding the number agreeing with it on line B, and against it on A, read the square measure of the board in feet. Measuring timber, square or round. 1. Take the length of the timber in feet and inches. 188 2. Find the number agreeing with the number of feet and inches on the hne C, and sHde it to the figure 12 on D. 3. Read-off the cubic feet hj finding a quarter of the girt in inches and finding a number agreeing with the inches of the quarter girt on the line D, and against it on C, read the cubic contents of the timber in feet. For sections from 211 to 214. Illustration of the Calculation of Ordinates. B C the base (100 feet) A B and A C the two equal side& (300 feet each) of an isosceles triangle. A, the angle at the apex of the isosceles triangle (20°) being double the angle formed by the deflexion of the chord line B C from the tangent line B a (10°.) The angle at A is at the centre of the circle, of which B 10 C is an arc. E D is a middle portion of the horizontal diameter of the circle ; which, in its whole length, is double the radii A B and A C = 600 feet. E D being equal to the chord line of the given arc (100 feet) it has 50 feet on each side of the centre A. There- fore the distance from D to the end of the diameter in the direction of E, is 350 feet, and the remainder in the opposite direction is 250 feet. By a known principle in mathematics, if 350 is multiplied by 250, and the square root of the product extracted, it will give the length of the ordinate D C. In the same manner the line E B is found. Shorten the side in the direction of E, 5 feet to c, leaving that line 345 feet, and making the other 255, and intermultiply them and extract the square root as before, you obtain the ordinate c n. Subtract the standing ordinate D C from the ordinate c 7i, and the remainder will be that part of the ordinate which is above the chord line B C. Proceed in the same manner with v s, and all other m'easures on the diameter line E D, moving 5 feet at a time along said line. In this manner all the offset lines (called ordinates) above the said chord hne, are obtained, as far as the middle ordinate A W. Then by ktverting their order, all between W and B may be set down. In this manner the table under section 210 was made. 189 190 For sections 224, 225 and 226. In taking the cross areas of excavations and enibanhnents, it is generally preferable, when very irregular, to suppose the base and surface level, and of course, parallel, however uneven they may be in reality. Then add their calculated lengths, halve their sum, and multiply that by the distance between their levels. This imaginary trapezoidal result is then to be reduced to the truth, by casting and subtracting the vacant places. Examvle, Fig 1. A B is a side hill. Cast the trapezoid B C E G in the usual way. Cast the triangle A B C in the usual way, when a triangle is made between two parallel lines, A D and B L. Ur the whole trapezoid, A D E G, may be cast, and the triangle AD B be deducted. 0^ In all cases the distance between parallel lines, and between an apex and a base, the difference of levels is the dis- Examvle Fig. 2. Z T O m N H is the uneven surface of a piece of required'excavation. Add Z X and R W, halve their sum and multiply that by the distance between the ascertained level ot the base R W, and the level of the highest point to be excavated, Z A ; then cast and subtract the vacant spaces. First, cast the trapezoid Z X T S, in the usual way. Second, cast triangle HNS, whose base is calculated from the central hench (fix^ed stake) and whose perpendicular is the difference in level between the apex at H and the base N. Third, consider the figure T N O t., as a trapezoid, and calculate it as such; for though the side O uis not parallel io T N, it is a case which may be averaged by the levelled hne O M being made to be intermediate in height between the true levels of O and u. Fourth, add the areas of these two trapezoids and the triangle, and subtract their sum from the factitious area of the whole assumed trapezoid. Remark. This transverse area may also be cast, directly, by casting the trapezoid R W K H-then the trapezoid K H V r- then the triangle V O Z-then, last, make an average triangle u r N. But the point would require a special measure, and both ends of the base line u r, would require more time and become more comph- cated, than by adopting the deduction method. In all cases the sur- 191 face is most accessible ; consequently may generally be easiest measured. Besides, the engineer can always construct his general ideal trapezoid in perfection, and fix some points of it to his benches. Power to overcome friction in a flouring mill, from sec. 315, con- tinued. Suspend weights on a water-wheel at its periphery on a horizon- tal level with its axis, cog-wheel, or other vertical wheel, until the whole gearing, stone, &c., start. Then calculate the wheel and axil power, so as to compare the advantage at the point of the appli- cation of the weight, with the point (or« average point) where the water acts on the wheel, whether overshot, undershot, or horizontal. INDEX Pages. 131, 132 85 76 10 to 25 56, 57 134 80 90 172 to 176 117 to 125 133 13 156, and on. 136, 137 172 Convexity of the earth, 111 Contraction of the vein, 143, 182 Curves, 104 to 112, 185 Accelerating forces, Architecture, Areometer of Baum, Arithmetic, Ascent and descent, Atmospheric height, Atmospheric pressure. Barometer, Calculations of examples Canals, — formula, Characters, Classes of strata Clouds, Coal measure, running, comparing. Decimals Density of materials for con struction, Displacement of water, Dynamics, Ellipse, Embankments, Evans' directions. Excavations, Falling bodies, Field book, Flumes, Friction of carriages, — of mills. 100 111 14 168 117 74 111 113, 190 100, 115 113,190 75 58, 69, 97, 98, 115 142 Funicular power, Genicular power. Geology, alphabet, series or 129 191 83 83 152 158 82 83 83 45. 179 72', 184 55, 56 78 Gonatous power, funicular, genicular, Guaging, Harbor survey. Heights and distances. Hydrodynamics, Hydrostatics, 77 Iron materials, 167 Jupiter's moons for longitude, 93 Kendall's tangent scale, 50 Land surveying, 47 to 73 Latitude, 92 Locks, 121, 122 filling and emptying, 124 Longitude, 93, 94 Materials for construction, 148 to 168 Mechanical powers, 84 Mensuration, 42, 177 Pages. Mills, 143, 144, 145, 146, 147 Millstones, 147, 148 trituration, ' 147 Moon's distance, under sec. 44, note, 34 Moving bodies on water, 117, 118 Natural Sines, Notation, Offsets, Ordinates, Pipes, Poestenkill mills, Pump, Bail-road, survey, extemporaneous preliminary, definite. Random lines, Retarding forces, Roads in general, surveying. Roots, square, cube, high powers, Rule of three, Sargeant's directions. Scale, reducing, raising. Shades for roads. Sines, natural, abridged, Single triangle. Sliding Rule, Specific gravity, 29 11 54, 180 108,188 132,133 143 81 89 to 116 89, 92, 95 , 89 95 98 55, 180 131, 132 125 68 19 21 24 17 95 to 100 60 62 127 19 viii and 31 62,182 187 76 of materials for construction, 168 Starting point in staking a curve, 105 Statics, 74 Supply of water for a canal, 123 for a mill, 138 Surveying, 47, 89, 92, 95, 100, 180 Table of logarithms, of refraction, Timber materials. Topography, Trapezoid, Tiigonometry, Useful rocks, , Velocity, Water as an agent. its powers, under pressure. works. Weirs, or water-pitches, table for it, Wood-cut figures, 169 170 165 148 64, 183 26, 32, 178 162 75, 138 to 140 75 137 80 131 145, 147 141 177 to 191 *X 6"^ rV V - ■V- o 0^ .Oo '^^ '^'%\- .A- S'''^. Deacidified using the Bookkeeper process. 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