for Engineering and other Purposes. 43 



cated surface of the internal flues or fire surface of the boiler, 

 by the quadratic scales 5 and 4. 



The new and old cylinders being in simple height as 15 to 

 14, the lengths of the beam, rods, arms, &c. would be ex- 

 tended by the scales for lines 15 and 14, and so on. If the 

 proportions of the vessels were altered, the application of the 

 scales would be modified as already shown by the diagrams. 



This paper might have been easily extended by other ex- 

 amples of many kinds, but I trust that enough has been said 

 to explain the principle and general method employed, that 

 alone being required in this place. 



The scale of geometrical equivalents in some respects re- 

 sembles the slide rule, but it possesses certain advantages. 



1st. It is easier to graduate scales of equal parts than those 

 in which the divisions are unequal. 



2nd. It is easier to subdivide equal divisions by the eye than 

 unequal ones, and they admit of the employment of the ver- 

 nier if required. 



Srd. The results are obtained in groups, whereas each se- 

 parate question requires a distinct setting with the slide rule. 



4th. In using the slide rule we are subject to two sources 

 of error, imperfect setting as to the gauge-point, and imper- 

 fect reading off' in the quantities. From the construction of 

 the scale of equivalents, as the scales all begin from one com- 

 mon line, the setting can never be disturbed. It is therefore 

 only liable to the single error of reading off', the settings being 

 constant. 



5th. In order to obtain the most accurate results it is occa- 

 sionally desirable to double or halve the quantities, so as to 

 get as far from the zero of the scales as possible, whereby the 

 error of reading will be proportionably diminished. The 

 very nature of the slide rule limits this application, as some- 

 times only a third part or less of the scales remain in contact. 



6th. But the most serious drawback to the slide rule is the 

 constant practice that is required for using it with facility, 

 arising principally from the difficulty of assigning the true 

 value to the answers, whether units, tens, hundreds, thousands, 

 &c., and the constant reference to the tables of the gauge- 

 points. 



Now, as in all cases the true values are expressed upon the 

 scales of the new instrument, and their particular names de- 

 note at the same time the functions and relationships of the 

 several scales, both inconveniences are in a great degree, if 

 not entirely, obviated. 



7th. It is true, the slide rule may be employed for any arbi- 

 trary setting, whereas the present instrument is in a measure 



