374 Proceedings of Philosophical Societies. [Not. 



amplitude, and module is the object which he investigates 

 in this continuation of his researches. " This may be obtained, 

 either by means of a table constructed for each given value 

 of the angle of the module, or by means of a system of 

 tables, which shall be constructed according to the variation, 

 by equal and sufficiently small intervals of the amplitude and the 

 angle of the module." The author discusses the advantages and 

 -difficulties of each of these methods. The second supposes a 

 task, which it would take a long time to execute. To lessen 

 in some measure the difficulty, his tables 8 and 9 offer a prepara- 

 tory labour to calculators, which may also partly supply the 

 want of more extensive tables ; but as they proceed only from 

 one degree to another, either of the amplitude or of the angle of 

 the module, their interpolation will necessarily be more difficult, 

 or less exact than if these intervals were smaller. 



In order to avoid double interpolations, it would be necessary 

 to return to the first method ; but the calculation of this table 

 would be so long that there ought to be a great number of 

 functions to calculate upon the same module, to induce one to 

 undertake so considerable a preliminary work : to attain the 

 same end with greater ease, the author shows that a table of only 

 a few lines, formed upon a given module, will serve to calculate, 

 as far as 10 decimal places, or more, the functions E and F 

 correspondent to any value of the amplitude <p ; and that it will 

 be sufficient for this purpose to add to the common mode of 

 calculating the interpolation, that of some very easy trigonome- 

 trical formulae. This method may be rendered still more simple, 

 if the calculation is confined to seven decimal places ; but it is 

 explained in detail, and applied to examples, with the utmost 

 care, in order that the exactness of the results may be warranted 

 as far as the fourteenth decimal place. This precision may 

 probably never be required : it is the utmost limit of exactness 

 which can be obtained by the tables now known. The tables of 

 the logarithms of numbers, by means of some artifice in calcu- 

 lation, would show it to the 20th or 22d decimal places ; but 

 beyond that number, common arithmetical calculations must be 

 resorted to, by which alone, an indefinite degree of exactness 

 may be obtained. 



Such is in substance the introduction placed by the author at 

 the beginning of a work, of which it is impossible for us to give 

 a neater or more complete idea. The reader will find it rich in 

 formulae, in acute developments, and in subsidiary tables scrupu- 

 lously calculated, some to 10, and others to 14 places of 

 decimals, with differences as far as the third or fourth order. 



History of the Astronomy of the Middle Ages, pp. 700, 4to. 

 with 17 plates, by M. Delambre.— The author calls the middle 

 age of astronomy the interval between the period when the 

 Greeks ceasing to write had their place supplied by the Arabians, 

 Persians, and Tartars, and that, when Copernicus, restoring to 



