224 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[July, 



if it were possible to contrive and construct a system of machinery 

 by which the force required could be accomplished and applied, the 

 violence with which it would have to operate in order to effectuate 

 its purpose, (admitting its practicability,) would be such as to pre- 

 clude the possibility of its employment. 



To the balloon then, as affording the legitimate and only probable 

 means of grappling with these difficulties, are we naturally referred : 

 and that, by its means, success could be attained to such an extent, 

 as to satisfy the scruples of the most incredulous, we are fully pre- 

 pared to prove; not by the proposal of any crude and complicated 

 contrivance, the operations and effects of which are neither defined 

 nor definable, neither adjusted by the strict laws of science, nor re- 

 concilable with practice, but by a mode at once simple in the extreme, 

 susceptible of an examination as critical and close as that with which 

 we have been sifting the scheme before us, and above all, answering 

 the obligations imposed upon it by means so far within the limit of 

 its attainments, as to leave no doubt as to its ultimate success, in some 

 degree corresponding with the purposes which it might be expected 

 to subserve. This, should our readers still continue to feel an interest 

 in the subject, we may take an opportunity of laying before them 

 upon a future occasion. 



A SIMPLE METHOD OF COMPUTING THE SPHERICAL 

 EXCESS, WITHOUT THE AID OF LOGARITHMS. 



By Oliver Byrne, late Professor of Mathematics at the College for 

 Civil Engineers ; author of " The new and improved System of Lo- 

 garithms," " The Doctrine of Proportion," " The Practical, Complete 

 and Correct Ganger," Sec. 



Although Professor Dalby's rule for calculating the spherical excess 

 is simple, yet in many instances it is inconvenient, especially when 

 logarithms are not at hand, or when the tables are limited ; moreover, 

 when the radius of curvature at the place of observation is greater or 

 less than 365154-6 X 57-2957795 feet, a more appropriate constant 

 logarithm than that given by Dalby might be used. However, it will 

 be found in practice, that the spherical excess can be determined in all 

 cases, by the method here explained, in one half the time usually em- 

 ployed to find the number corresponding to the difference of the lo- 

 garithms mentioned by Dalby, (of which we shall speak by-and-bye,) 

 and therefore the trigonometrical surveyor will in this particular cal- 

 culation economise one half his time, at least. 



The following table will be sufficient from 90" down to -j-^Vj of a se- 

 cond, and can be increased or decreased at pleasure by multiplying or 

 dividing by 10, if this range be not extensive enough; the change is 

 easily effected by increasing the number of perpendicular lines to the 

 right and left of those already drawn. Such an alteration may be ne- 

 cessary in speculative inquiries, but can never be required in practice. 

 This method of calculating the spherical excess, which depends on the 

 succeeding table, may be better explained by a few examples than by 

 a written explanation. 



Examples. 



I. The area of a triangle is 1764724375 square feet; what is the 

 spherical excess in seconds ? 



Area 1764721375 



Nearest in the table .. 1697710640 -8-- headed tenths. 



Remainder 



Next nearest in the table 



Remainder 



Next from the table 



67013735 

 63664149 



3349586 

 2122138 



•3* headed hundredths. 



.1 headed thousandths. 



Therefore the spherical excess : 



&c. 



0"-831.. 



•831 



II. Suppose that in the trigonometrical survey of Ireland, under the 

 superintendence of Colonel Colby, the county of Wicklow, which 

 occupies 2 19SG4065SS square feet, to be taken up with one large 

 triangle, what would be the spherical excess. 



219864 '.n;,".SS Area in square feet. 

 10-0000 .. 21221383006 which terminate in the column headed 

 tens of seconds, opposite ]. 



•3000 



•0600 



765113582 

 636641490 



126472092 



127328298 



which terminate in the column headed 

 tenths of seconds, opposite 3. 



answering to six hundredths of a se- 

 cond. 

 1H37M4 

 •0005 .. 10610691 there are no thousandths of seconds, 



i but 5 ten thousandths, found by 



10"-3605spl. excess. &c. curtailing another figure. 



in. The lengths of the chords of a spherical triangle on the surface 

 of the earth are 13, 14, and 15 miles respectively ; the area of the 

 triangle formed by these chords is 23417S56O0 square feet, (which 

 may be taken as the area of the corresponding spherical one, on 

 account of the sides being very small compared with the radius of 

 the earth.) The three angles of the plane triangle are 53° 7' 

 48 '-358, 59=29' 23 '-136, and 67 22 48"«606, required the sphe- 

 rical angles. 



23417- 



1-000 .. .. 2122138300 .. for 1 second. 



•100 



•003 



219647300 

 212213830 



7433470 

 6366114 



, , for -pj of a second. 

 ■ • for TTfin; of a second 



1-103 seconds the spherical excess. 



One third of the spherical excess must be added to each of the plane 



triangles to obtain the spherical angles. Therefore 1"-103 divided 



by three = "*36S nearly, hence we have, 

 Plane angles 53° 7' 48"-358 59" 29' 23 "-136 67° 22' 48"-506 



Third of exs 



•368 



•368 



53 7 48-726 59 29 23-504 67 22 48-874 

 In this exa mple the excess is distributed equally among the angles, 

 because there is no reason to believe one of them to be more erro- 

 neous than another. But if one of the angles be suspected to be less 

 correct, or less to be depended on than the others, to this angle must 

 be applied a greater or less portion of the whole correction, according 

 as it is thought to be in excess or defect. 



Dalby's Rule. — From the logarithm of the area of the triangle taken 

 as a plane one, in feet, substract the constant logarithm 9-3267737, 

 and the remainder will be the logarithm of the excess above 180°, in 

 seconds, nearly. By this rule let us work the second example just 

 given:— 



