INT 



Application Or we may arrange the expression thus, 



175 



I N T 



ml under either form, we have the value of x, which 

 corresponds to the lea-,t, or greatest, value of y. 



Dr. Halley, in the Phil. Trantt. for the beginning of 

 the year 1695, resolved Walttier's observations into two 

 set*, viz. 



June 2. 45V June 8. 44975. 



12. 44883. 

 16. 44990. 16. 44990. 



From the first set, if we reckon the time from June 

 2, we have, 



.r, = 7 y, = 4*934 



* s = 14 y, =44990 



and hence, by the second formula, 



_g8 X 533-7x477 

 '- 4x533-2x177- 



To this, add the two days in June, preceding the 

 first observation, and we have ll 4 20" 1 - !" O. S. for 

 the time of the solstice. 



According to the second set of observations, reckon- 

 ing from June 8, we have 



y = y = 4497-i 



r, = 4 y, = 41883 



*, = 8 y t = 44990 



Hence, by substituting in the second formula, 



This period, added to 8th June, gives 11 L 20 h - 23' 

 fur the time of the . .tfering from the former 



determination by only 22 minute*. 



Another application o/tke Formula. 



Gassendus, at Marseilles, in the year 1636, observed 

 the summer solstice, by a Gnomon 55 feet in height, 

 and be found, that of such parts aa the gnomon con- 

 Uined 89128, 



June 10. N. S. the shadow was 31 766 

 Jane 20. 31753 



June 21. 31* 



June 28. 31759. 



These being divided into two sets, vi. the 19th, 

 SOth, 2id ; and 1 9th, 21st, 22d, we have, 



in Set. W StL 



*o=0 jr = 31766 * =0 y =31766 



*,=! jr. =3 *, =8 y, =31751 



= 3 $., =31759 jr,=s y, =31759 



The first give* *==! 17*. 



y^ 



I :.. 



= -^-1-17-25- 



So that we may conclude the moment of the solstice 

 t/> have been June 20*- 17*- MP- in the meridian of 

 Marseilles. Thee conclusions, deduced from real ob- 

 -- have been of great importance in 



The formulae, by which we have determined the sol- Interval*, 

 slices, will equally apply to other cases, in which the "^ "V~" 

 values of y at equal distances from its maximum or mi- 

 nimum value are equal. 



We shall have occasion to advert to this subject again 

 under the article SERIES. (f) 



INTERVAL in Music, is the distance of two sounds, 

 as to acuteness and graveness ; what, however, is here 

 called distance, is, as Dr. Robison has observed, pure- 

 ly figurative and analogical, and not real : but the ana- 

 logy is very good, and the observation of it has led to 

 the discovery of precise measures of the intervals be- 

 tween defined or musical sounds. These last are such, 

 as preserve for a sufficient period of observation, or 

 comparison with other sounds, the same identical pitch, 

 or degree of acuteness. The multiplied and greatly 

 varied experiments of philosophers have shewn, that 

 this stability of pitch it accompanied by, and indeed oc- 

 casioned by the pulses or vibrations of some elastic bo- 

 dy, repeated at very quick and exactly equal intervals 

 of time. 



From the pitches of two given sounds, or the num- 

 bers of their VIBRATIONS (see that article and CONCERT 

 PITCH,) in a given short interval of time, as one second 

 for instance, the interval between them is therefore to be 

 somehow measured ; but it will be found, that this can- 

 not be done by considering the vibrations as lineal, and 

 taking their simple difference by subtraction as the 

 measure of the interval, because in this way all sorts of 

 absurdities or disagreements with the most simple and 

 obvious experiments, would follow in different cases. 



But if we consider each particular pitch or velocity 

 of vibration to be a togarilkm, or measure of a ratio, in- 

 stead of a lineal dimennion, or mere numeral quantity, 

 we shall then find, that the differences of these loga- 

 rithms, which are themselves also logarithms, naturally 

 and correctly represent the intervals of sounds. Every 

 interval, therefore, may be conidered as the modules 

 or unit of a particular logarithmic scale, as has been 

 shewn under the articles BINARY. COMMON, and HY- 

 PERBOLIC Logaritl.mi ; and every logarithm, of what- 

 ever species, a the measure or representation of some 

 interval of sound. 



Since the numbers of vibrations of simple elastic 

 strings of equal magnitudes, densities, and tensions, 

 art- tuund to be in the direct ratio of their length, it 

 follows, that the ratio* of these lengths of vibrating 

 strings are also correct representations of the intervals 

 yielded by the vibrations of these strings. And this, 

 although rather n unnatural and forced way of consi- 

 dering the measure, and effecting the calculation of 

 masical intervals, was the earliest, and continued 

 through several centuries to be the only mode which 

 mathematical musicians had, of representing and calcu- 

 lating intervals ; because in those days logarithms were 

 unknown, and the intervals then known (even the 

 Comma, having the ratio \, , the least of them) were too 

 considerable in magnitude, to be used in the conveni- 

 ent or natural representation of other intervals. 



But within a few years past, since the labours of the 

 late Mr. Marmaduke Fverard and others have brought 

 to light and shewn the relations of several intervals very 

 many times tmaUer than the comma, or least interval of 

 the ancients, Mr. Farcy has been enabled to adopt a 

 notation, which in terms of three (at the most, and of- 

 ten of two, ) of these very small intervals, is calculated 

 in a correct and natural way without negative signs, 

 (except in a few and unimportant instances,) to repre- 

 sent all musical intervals whatever, as lias been shewn 



