IS 47.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



243 



much time and Ial)Our in search of a simple and direct mode of calculating 

 logarithms, and though wholly unsuccessful, or very nearly so, as respects 

 the ostensible object of the inquiry, they have been rewarded by the dis- 

 covery of those interesting and momentous formulas which constitute what 

 is at present termed " the Theory of Logarithms." It is also worthy of 

 remark, that Briggs, Halley, Sharp, Vlacq, and others, who brought the 

 doctrine of logarithms to perfection, were not averse to arithmetical calcu- 

 lations; hut our modern mathematicians depend by far too much on purely 

 algebraical expressions, foreign' translations, and mere hocus pocus opera- 

 tions on operatical symbols. 



In an inquiry on logarithms, it is usual to put N = any given number, 

 a = the base of any system, and M = the modulus of the system. Substi- 

 tuting 1 + n for N, &c., we have 



log. (1-t-n) = M(n-Jn= + 5?!'-Jn* + ^n''-, &c., for the fun- 

 damental expression, from which several other formulae are derived, hitherto 

 used in the computation of logarithms. But the above series is only useful 

 when n is a very small fraction; while the majority of those deduced from 

 it, are only available in the process of determining logarithms from the 

 combinations of others. The value of M, in the above series, cost Mr. 

 Briggs 54 successive extractions of the square root, and 54 multiplications ; 

 and although many ingenious contrivances have been devised to abridge the 

 labour of these extractions, the process is at best very tedious. 



Lagrange converted the above series into 



log. m 



^M^ (.«'.- 1) - ^'"'-') ' + ('"•-')^ - &C.1 



by substituting m- for 1 +n ; r being entirely arbitrary. This formula can 

 be rendered as convergent as we please, and therefore the value of r can be 

 80 assumed, that the logarithm of any number, m, can be determined to 

 a limited extent, by using only the first term of the series, viz. fiom the 

 equation — 



log. m = r M (>«' — I). 

 This method, undoubtedly, is always applicable to the direct computation of 

 a logarithm ; yet it is the same in effect as that proposed by Briggs, and is 

 equally laborious, on account of the great number of extractions generally 

 required. 



It is, perhaps, unnecessary to dwell at any great length on the difficulties 

 attending the computation of logarithms by a direct process, independently 

 of other logarithms ; however, we cannot conclude these remarks without 

 giving a remarkable expression, deduced by Professor Wallace, of Edinburgh. 

 The form is this — 



log. X = — . . _' 



•1) 



X" »!(4" 1) 



in which m and n are any numbers chosen at pleasure; z, always some value 

 between and 1 ; and A, the given base of the system. This expression 

 leaves the base unrestricted, involves no infinite quantity, and is said by some 

 to be " of great analytical elegance ;" — yet, it is purely algebraical, and as 

 to its practical utility in the actual determination of a logarithm, it is just 

 as much use as any other intelligible hieroglyphics. 



Perhaps you will allow me to state a fact, which you have tested* — i. e. 

 that I have discovered a method by which the logarithm of any number, to 

 almost any extent, may be calculated, independently of other logarithms, in 

 a few minutes. Mathematicians and the curious will, I have no doubt, be 

 obliged to you for publishing the following results. It is well known that 

 when the diameter of a circle is one, the circumference is 



3-1415926o35897932384620433832795028841971C939937511, 

 to 50 places of decimals. Now, I find the logarithm of this number to be 



•497149872G94133S543al26828829089S873651G7832438044, which is 

 true to 50 places. For the information of the general reader, it may be 

 necessary to mention, that the logarithm of a number consisting of so many 

 places of figures, has not been before computed to anything near this 

 extent ; for, by any of the known methods, such a calculation is almost 

 impossible. From the above result, the logarithm of the area of a circle, 

 when the diameter is uniti/, may be readily deduced, and is found to be 



f-895089881366I7146392379049884191282011529856145642; correct to 

 the last figure. 



* [We have witnessed Mr. Byrne's facility in calculating logarithms without the use of 

 any book. It is highly desirable that his system of calculation should be revealed to the 

 public.]— Ed. C. E. & A. Journal. 



M'ith equal facility, we obtain the logarithm of the contents of a sphere, 

 when the diameter is unity, to be 



i-718998G2231049022I84250149021 129053768335957872764. 



M = -4342944819032»18276511289189166050822943970058036666. 



July, 18J7. Oliver Byrne. 



WARNER'S LONG RANGE. 

 For the following calculations of the dimensions of the balloons which 

 would be required for the purposes of Mr. Warner's Long Range, we are in- 

 debted to the courtesy of Sir Howard Douglas, whose scientific researches 

 have so greatly tended to disabuse the public mind of errors respecting the 

 resuscitation of an old project for aeronautic warfare. 



It has been already explained that Mr. Warner's apparatus consists of a 

 balloon, from which, when it has attained a proper altitude and position, 

 heavy shot or shells are to be let fall, being detached from the car by self- 

 acting mechanism : these missiles derive their destructive effects from the 

 velocity acquired by the action of gravity during their descent, or from the 

 disruptive force of an explosive composition contained in them. 



First of all, let it be required to determine the greatest possible velocity 

 which the shots will acquire. 



Falling bodies are acted on by two vertical forces during their descent 



the accelerating force of gravity, and the retarding force of the resistance of 

 the air. The former of these forces is constant at all velocities; the latter 

 increases very rapidly with the velocity, and may be assumed to vary as the 

 square of it. Consequently, the resistance to the progress of the balls be- 

 comes greater and greater, till at last it just counterbalances the action of 

 gravity : in this stage of the descent, the velocity is said to have acquired its 

 "terminal value," beyond which further acceleration is impossible. When 

 once, therefore, a falling bodyhas acquired its terramal velocity, it is no longer 

 accelerated, but continues its descent with precisely the same uniform velocity 

 (unless new forces are brought into operation), till it reach the earth. 



Now it appears from numerous experiments, that the terminal velocity of 

 a 12 lb. shot, filled with lead, (that is, the greatest vertical velocity which the 

 shot can acquire by descent) is 419-6 feet in a second : and to acquire such a 

 velocity the ball must fall from a height of not less than 27492 feet. These 

 results may be safely relied on, as they express the mean of a vast number of 

 experiments. The terminal velocities of solid shot of various sizes differ 

 considerably. As the solid contents of spheres vary as the cubes of their 

 radii, and their surfaces only as the squares of their radii, it follows that the 

 larger the shot the heavier will it be in proportion to the surface exposed to 

 the air's resistance, and therefore the greater will be the terminal velocitv 

 For Shells filled with an explosive composition the terminal velocity is less 

 than for solid shells of equal size, the former being lighter in proportion to 

 the surface exposed to the resistance of the air. 



If the resistance be taken to vary as the surface and the square of the 

 velocity conjointly (the surface varying as the square, and tlie weight as 

 the cube, of the radius), it may be easily shown that the terminal velocity 

 varies as the root of the radius. Heuce, r = 178 V d is a general expres- 

 sion fur the terminal velocity of a ball of d diameter, the constant 178 

 being determined by numerous experiments. 



The doctrine of terminal velocities is beautifully illustrated in the de- 

 scent of the parachute, which, after it has attained a certain velocity, will, 

 if properly constructed, conlinue to descend uniformly, without any further 

 acceleration. Another admirable illustration is aflbrded by falling rain 

 which, unless retarded by the air, would be so much accelerated as to 

 destroy vegetation. 



The idea of defence of fortified places by " vertical fire" — that is, by 

 shot discharged so as to fall nearly vertically on the heads of the be- 

 siegers — was promulgated by the celebrated mathematician, M. Carnot, 

 who, however, totally overlooked the resistance of the air, and supposed 

 the shot to describe parabolas. lu a Reply* to his theories, it was shown 

 theoretically, that the relardation of shot desceuding vertically would 

 render them all but inoperative ; and the theory was confirmed by actual 

 experiments, undertaken by the author for the especial purpose of testing 

 its accuracy. The following extract details the nature and results of 

 these experiments : — 



* " Observations on the motives, errors, and tendency of M. Carnot's priDciplea of 

 defeoce, showing the defects of his new system of fortification, and of the alterations he 

 has proposed with a view to improve the defence of existing places. By Colonel Sir 

 Howard Douglas, Bart., K.S.C. C.B. F.R.S., Inspector-General of the Koyal Military 

 College. London : printed for T. Egerton, bookseller to the Ordnance, Military Librarv 

 Whitehall, 1819." ' 



33* 



