Nov. 7, 1884.] 



♦ KNOWLEDGE ♦ 



383 



A. We have had a good deal of trouble over our preli- 

 minary inquiry into arcs. Can plane surfaces bounded by 

 curved lines, and solids bounded by curved surfaces be 

 more easily dealt with '! 



J/. Much more easily. We will take them next ; and 

 consider also the more difficult case of curved surfaces. 

 After that preliminary inquiry we will consider a number 

 of cases of interest, in the measurement of various curves, 

 surfaces bounded by curves, volumes, curved surfaces, and 

 so forth, taking very simple cases first. 



A. But will you not give me some examples of measuring 

 arcs. 



M. I must first explain how, instead of the chords and 

 tangents shown in Fig. 3 we have usually to employ other 

 approximations to the length of curved arcs. 



(To ie continued.) 



GBAPniCAL PROJECTION OF AN 

 ECLIPSE OF THE MOON. 



IN compliance with a request preferred by several sub- 

 scribers, we propose to explain the method of graphi- 

 cally projecting a Lunar Eclipse from the details given in 

 the Nautical A Imaiiac : and, in order to render the prin- 

 ciple perfectly intelligible, we have given all the successive 

 steps (including the simple calculations) in the production 

 of the subjoined diagram of the recent eclipse of Oct. 4, 

 which has attracted so much attention. 



First let us see what the Nautical Almanac furnishes us 

 with. Turning to page 401, we find : — 



Elements. 



tireenwieh Mean Time of Opposition in R.A., h. m. s. 



Oct. 4 10 8 5-1 



iWoon's Right Ascension 44 2502 



o / // 



Moon's Declination N4 57 57"9 



Sun's Declination S 4 46 33-6 



Moon's Hourly Motion in R.A 34 135 



Sun's Hourly Motion in R.A 2 167 



Moon's Hourly Motion in Declination N 10 52'9 



iiun's Hourly Motion in Declination S 57'7 



Moon's Equatorial Horizontal Parallax .59 23-0 



Sun's Equatorial Horizontal Parallax 8'9 



Moon's 'True Semi-diameter 16 125 



Stm's True Semi-diameter 16 2'4 



V 



of the earth's shadow ^ Moon's horizontal parallax + Sun's 

 horizontal parallax - Sun's semi-diameter — i.e., in this case, 

 5° 9' 17"-1 -i-8"-9 - 16' 2"-4 or 43' ■2,3"-6. To this we must 

 add l-60th, say, 43"-4, on account of the earth's atmosphere, 

 and we shall finally get 44' 7" for the semi-diameter of the 

 shadow. Take, now, any scale of equal parts (one of 40 to 

 the inch, or IJ inch to 1°, was employed in constructing 

 our diagram above), and with 441 of them as radius, from 

 centre C describe the circle A D B R. This will represent 

 a section of the shadow of the earth at the Moon's distance. 

 Draw the diameter A C B to represent a parallel to the 

 equator, and make CG perpendicular to it= 11' 24"3, the 

 difference between the Moon's declination and that of the 

 centre of the shadow. We must be careful to take G above 

 0, because the Jloon's centre is north of that of the shadow. 

 The beginner will note that, as the Sun's declination is 

 south, that of the earth's shadow must be north. Taking 

 the hourly motion of the Sun in R A from that of the 

 Moon (2' lG"-7 from 34' 13" 5) we get 31' 56"-8 or 1916"-8 

 as the hourly motion of the Moon in R A from the sun, 

 which we may now consider a fixture ; but it must first be 

 reduced to the arc of a great circle by multiplying by the 

 cosine of the Moon's declination. This is done in three lines 

 by logarithms. Thus : — 



1916"-8 log. 3-282.5768 

 4" 57' 57" -9 cos. 9 9983668 



1909"-6 or 31' 49"-6 



3 280943G 



B^Now then to begin our figure. 



First, the figure of the earth is a spheroid, and not a 

 sphere, so that her shadow is not rigidly circular. Hence, 

 to have a mean radius, we will reduce the Moon's horizontal 

 parallax to a latitude of 45''. This reduction we find to be, 

 in the proper tables, 5" -9, so from 59'23"0 we take 5"-9, 

 leaving the reduced parallax 59 ' 17"-1. The semi-diameter 



So we make 0=31' 49" -6, and C P, perpendicular to it, 

 = 9' 55"'2, the Moon's hourly motion from the centre of 

 the shadow in declination, observing to place P above 0, 

 because the Moon was going northward with reference to 

 the shadow. Join O P, and parallel to it, through G, draw 

 the line N G L. This gives us the Moon's path with 

 respect to the shadow. On N L let the perpendicular 

 OK fall. Now, at lOh. 8m. 5"ls. the Moon's centre was 

 at G. To find where it was at lOh. (marked X on 

 NGL) we say as 60m. :8m. 5-ls. ::0P toGX. We take 

 the corresponding measure ofi" our scale, and, extending it 

 from G, so find X. Then, in a perfectly obvious way we 

 set oft" the distance P on either side of our 10 o'clock 

 point, and sub-divide these hours so graduated as minutely 

 as we choose. Those in our figure might be so divided into 

 60 mins. each, without difiiculty. Then will the times 



corresponding to E K and 

 H represent the beginning, 

 middle, and end of the 

 eclipse respectively. Cir- 

 cles described round E K 

 and H, with a radius = 

 the Moon's semi-diameter 

 (16' 12"-5), will show the 

 position of the Moon at 

 these instants. R T shows 

 what is known as the mag- 

 nitude of the eclipse. If, 

 with C as a centre, and a 

 radius = C B— the Moon's 

 semi-diameter, we describe 

 a circle cutting L N in S T, 

 the instants of the Moon's 

 reaching these points are those of the beginning and end 

 of total darkness. 



Erkatum.— In the reply to "W. H. K. S.," column 2, p. 373, 

 in our last number "wilful imbecility" should be "wilful un- 

 belief." 



