ISJ5.] 



TiiE CIVIL kngini':kr and architixt's journal. 



3! I 



Fig. 6. 



(3.) Tlie distance between tlie second and fourth of four stations, fig. C, 

 taken in tlie same plane, forming a trapezium, (as 

 in tlie next diagram,) togetlier with the angular 

 distances taken at the first and third stations, 

 measured from an unknown diagonal, heing given 

 to find the remaining linear and angular distances 

 connecting those stations. In this prohlem the 

 best position for the point 0, is the second or 

 fourth station. 



Let A he the 1st station, 1) the 2nd, C the 3rd, 

 the 4lh ; and the angle O A C = x, angle A C O 

 then will = 180°— a — i-a-, or e—\SO-a-/i, 

 e — 108° 15'. Having thus far premised we shall 

 now applv the general proposition.. 



' O A : B : : sin O B A • sin A. B : : sin c : sin (a + c) 



O B : C : : sin C 15 : sin B C : : sin (S ^- rf) : sin (/ 



C : A : : sin A C : sin O C A : : sin r : sin (e— x) 



.■ . sin c sin (* + (/) sin r = sin d sin (a + e) sin (e — .»). . • . cot .r = 



cot e+ cosec (a + c) sin (A + d) sin c cosec d cosec e. From which we liave 



the following practical logaritlimic 



Utile. — Jdd tor/el/ier the log cosecant of (a + c), the log sine of {h + d), the 

 log nine of c, the log cosecant of d, and the log cosecant of e . the natural 

 number corresponding to this sum, when a proper allowance is made in the 

 index, added to the natural cotangent of e, will give the natural cotangent ofx. 

 Given — Lineal distance from 1st to 3rd station, A C = 19712 feet = n. 

 Angular distances at 2nd station, A B = 39° 47' = a; B C = 31° 

 58' » 4. 



Angular distances at 4th station, A B = 25° 17' = c ; B C = 30° 

 25' = d. 



To find X. 

 log cosec (a + c) = log cosec 65° 4' =10-0424890 

 log sin (6 + d) = log sin 68° 23' = 9-9083285 

 log sin c = log sin 25° 17' = 9-6305243 



log cosec d = log cosec 30° 25' = 10-2264073 



log cosec e = log cosec 108° 25'= 10-02-24140 



Rejecting 50-49-8902231 

 we have 1.8902231, which corresponds to 0770040 of a natural number. 

 Now, the natural cotangent of e = nat cot of 108° 15' = nat tan of 18° 15' 

 0-3297505, but negative. 



Then, from 0-7706400 

 take 0-3217505 



Fig. 7. 



nat cot x= -4408155 hence 

 ^•■= 65° 55' 14". 

 M'hen this is known, the determination of the other linear and angular dis- 

 tances can present no difficulty. A - 13976-96 feet ; B = 29075-0 feet. 

 (4.) The distance between two stations, fig. 7, and the angular distance 

 taken at each of them, to two others on the 

 same side, being given when one of the sta- 

 tions is inside of the triangle formed by con- 

 necting the other three: — To find the other 

 angular and linear distances. 



Let \ he the 1st station, B the 2nd, C the 

 3rd, the 4th ; the linear distance from the 

 first to the second station 3000 feet ; the 

 angular distances at the first station, A B - 

 33° 49' = o, A C = 29° 45' = 4 ; the an- 

 gular distances at the second station, O B A = 

 30° 18' = c, B C = 45° 21' = d. 

 As before we find 

 cot X = cot e + sin a cosec 4 cosec c sin d cosec e 

 which gives the following 



Sule. — .^dd together the log sine of a, the log cosecant of b, the log cosecant 

 of c, the log sine of d, and the log cosecant of e ; the natural number corres- 

 ponding to this sum, rejecting 50 in the index, added to the natural cotangent 

 of e, will giie the natural cotangent ofx. 



log sin a = log sin 33° 49' = 9-7434943 

 log cosec 4 = log cosec 29° 45' = 10.3043288 

 log cosec c = log cosec .-iO' 18' = 10-2276086 

 log sin d = log sin 45° 21' ^ 9-8521218 

 log cosec e rzi log cosec 34° 47' := 10-2437636 



50-3733771 

 Cancelling 50, we have -3733771 



2-3625290 natural number corresponding to -373371; 1 

 1-4397048 



3-8022339 — nat col x, hence x = 14° 44' 7". 

 A 0, B, O C, can be readily found by the rules of plane trigonometry. 



ON THE PREPARATION OF LIME 



FOR FRESCO AND OTHER PURPOSES OF PAINTIKG AXD ARCUITECTURE, 

 AND THE DEFECTS OF PLASl-fiR KEYING. 



Without entering too minutely into the mere tradition:iI theories of 

 Pliny, and wliich were very consiilerably adopted by liis succosssors, 

 or following the moderns in their misa|iplicatious of chemical science, 

 we n)ay justly assert, that if no actual retrogradation lias attended the 

 various uses of lime no advances have been made or improvements 

 achieved during many hundred years. And, although 1 demur to the 

 iisii illy received encomiums lavished on Roiuan skill and Roman per- 

 f'<-lii)n in the manipulation of lime and formation of crmenl«, and 

 lirrnly believe their opinions to liave been erroneous, except with re- 

 ference to their own climate, and llieir laws for ! eepin? mortar three 

 yiais to have been the result of necessity not choice— that is to say, 

 instead of a general improvement and greater intensity of indurating 

 power having been acciuircd by sncli keeping, that the mortar so kept 

 must have been jialpably and extensively deteriorated, in other words, 

 carbonated or returned to the state of chalk, and therefore improved 

 only for their use, simply because the proportions of tlie elements — 

 lime, sand and water— as handed down to posterity, w-ere infinitely 

 loo strong, too fierce and too rapidly indurating fur'such a climate in 

 the first instance, as presumptively proved by the fact of Pliny's pro- 

 portions for a strong mortar being much richer than those used 

 by our masons in a damp atmosphere requiring double the strength; 

 and 1 am satisfied we mis-read Pliny as to the proportions, in conse- 

 quence of the masons of his day having carefully slaked all their lime, 

 as we do chalk lime, the day before, and that his measurement of 

 quantity referred to the slaked hydrate, not the dry caustic lime as 

 with us, still this supposed error rather corroborates than weakens the 

 presumptions that Roman inert ir, nhtn first mixed, was infinitely too 

 strong for Roman use, i. e. had far too much lime. 



Roman Proportions. 



1 part {or niie-thiid; river sand 



:.' parts of lime 

 -winth, if I am correct in believing this 

 f> mean hydr.tte of lime, is equal parts of 

 mo and sand. 



English Proportions. 



1 load of lime, which is pretty nearly two 

 of the hydrate 



2 loads of sharp sand 



-that is, equal parlsof hydiateof lime and 

 and. 



Be this, however, right or wrong, Roman mortar could never gain 

 strength by three years' keeping, and English mortar so kept, though 

 much pleasanter to the workmen, as being more plastic, would be good 

 for nothing. 



Higgins, the only practical English writer on the subject, appears 

 to have perplexed himself as much about carbonic acid, and its action 

 on cements, as M. Vicat the French writer, who absolutely made a 

 series of experiments to ascertain, to the breadth of a hair, the depth 

 to which it would penetrate a thin stratum of hydrate of lime, in so 

 many weeks, days, hours and seconds ; as if two facts were not obvious 

 to common sense, viz., 1st, that carbonic acid woulil return it to the 

 state of chalk, and must, in the nature of things, be the palpable de- 

 stroyer, not invigorator of cements, and 2ndly, that whether it could 

 penetrate such stratum of exposed hydrate to the di ptli of one line oi 

 fifty were equally immaterial to the practical man, for it never can be 

 supposed to enter apprecialilg the internal structure of a six /ett wall, 

 aud very insignificantly that of an eighteen inch one, and that too after 

 the setting of the mortar, when chemical action must be lessened in 

 proportion to its dry state. 



Every mason's man knows two facts, viz., that in direct proportion 

 to the sharpness of the sand and the rapidity and fury of tlie slaking 

 (technically called boiling), especially the slaking togetlier the lime 

 and sand by water, is the jiresent strength of ihe indurating pomer and 

 the fuiure firmness of the cement, — which enables us tn combat another 

 error arising out of the flippant applications of chemical theory by 

 bookmakers and elementary teachers, viz., the foolish supposition that 

 sand or silica acts chemically as an acid on hydrate of lime, than which 

 nothing can be more grossly absurd, for were this the fact silicate of 

 lime would be palpable in every old wall, and its original sand nearly, 

 if not entirely, invisible, whereas the reverse is the fact, and more- 

 over, if such wire-drawing science were correct, the finer and more 

 soluble the sand the belter the mortar, while a ])ure silica, or even 

 ground flints, would surpass all other forms for cement, which is pal- 

 pably a ridiculous, nay a proved error — for more than one worthy and 

 talented mt mber of society has been so far mislead by these smatter- 

 ing gentry as to waste time and money in forming compounds of no 

 value whatever to mankind ; indeed a patent has been very recently 

 taken out by an ingenious man' for making an artificial stone by mix- 

 ing solutions of a true silicate of potiis with ground flints, granite, 

 &.C., on these wire-drawn theories, and which must, in the nature of 

 things, prove a failure, when by abstracting the pseudo-science and 



