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Table VI.
°' Vln S what number of plain tiles, or pan-tiles, will covery ar ea from 1 to 10,000 feet.
Feet
Super.
1
2
3
4
5
6
7
89
10
20
SO
to
so
60
70
80
90
100
200
300
400
S00
600
700
800
900
1000
2000
3000
4000
3000
6000
7000
8000
9000
htooo
Plain tiles.
Pan-tiles.
Gauges.
Gauges.
6 inch.
6} inch
7 inch.
11 inch
12 inch
13 inch
n
7
8*
]§
14
14
15
14
13
3}
3
4
22}
21
194
5
45
30
28
26
6&
6
37^
35
32}
8j
7}
6f
45
42
39
10
9
8
52}
49
45}
J 1 !
104
9£
60
56
52
12
lOf
67*
63
584
15
134
12
75"
70
65
16f
15
134
150
140
130
33|
30
2<3
225
210
195
50
45
40
300
280
260 1
66}
60
53J
375
350
325
83}
75
66§
450
420
390
100
90
80
525
490
455
116}
105
934
600
560
520
133}
120
106}
675
630
585
150
135
120
750
700
650
166}
150
1334
1500
1400
1300
334
300
266}
2250
2100
1950
500
450
400
3000
2800
2600
666}
600
5334
3750
3500
3250
833}
750
666f
4500
4200
3900
1000
900
800
5250
4900
4550
1166}
1050
9334
6000
5600
5200
1333}
1200
1066}
6750
6300
5850
1500
1350
1200
7500
7000
6500
1666}
1500
13334
15000
14000
13000
33334
3000
2666}
22500
21000
19500
5000
4500
4000
30000
28000
26000
6666}
6000
53334
37500
35000
32500
83334
7500
6666}
45000
42000
39000
10000
9000
8000
52500
49000
45500
11666}.
10500
93334
60000
56000
52000
I33334
12000
10666}
67500
63000
58500
15000
13500
12000
75000
70000
65000
16666}
15000
133334
9?n,
e mi
Iti rj,”"' ^Pfanation of the Tables and their Construction.^>ch es > which may be calculated by a proportion, inb 6tl % tj 6 ^ le terms 13 constantly the same, and conse-. 6f s, are t 0t ' ler two only variable, the given data, or num-th *hw ' Va ^ s two kinds ; one of the kinds is placed in. e fetvitj x Un< ^ t ' le other in another row, at a right angleth^’ and * t ' le ^ ormer being parallel to the edges of thef 0 e s aid "P°n the left hand, and the other perpendicular toho^ er i th^-’ an< ^ at the top of the page : we shall call the0 y f|2 °ntal ^ lven vertical column ; and the latter, the giventhe <^° W ’ t ' ie first part of the given horizontal rowf 0 l> ; vertical column, contains the title of this
Md e aclr given numbers of this horizontal row
lih a the t‘n 0t ^ er * n succession, having each a separate cell* 0 With P e , at the top ; then other rows are made in a*fta>t the aCa ce H’ parallel to the given vertical column,a l s o i| res Peetive numbers which form the said columns\ 11111: the 6 fi le numbers in the left-hand vertical
ift atl 8we r „ 11 eac h of these numbers, so disposed, are to be>s gj v the two numbers at the extremity of the legsO the Co row ‘ From this it appears, that the answerH ic ,^h acou rse of the given column and row, tracedtft e 1 the f 1Ven datum parallel to the given columns in>* the p- ta are inserted. All the useful variations oft. 8 l ven species, or kinds, are expressed by one3 b
or several arithmetical progressions, in the given verticalcolumn, and the other species in an arithmetical progressionin the given horizontal row. If the given vertical columnconsist only of one arithmetical progression, the answer forany number in the said column, between the first and thelast, may be obtained at once . but if the given verticalcolumn consist of several arithmetical progressions, viz.from 1 to 10, from 10 to 100, from 100 to 1000, &c., thecommon difference of the following progression being 10times greater than that which precedes it, then the givennumber belonging to the vertical column must be dividedinto component parts, so as to correspond with as many ofthe numbers to be found in the progressions; each of theanswers are to be taken out separately, and added together,and the sum will be the answer in full. Where the givenvertical column consists of one or more arithmetical progres-sions, so will each vertical column of answers consist of asmany arithmetical progressions, and each horizontal row ofanswers will also form an arithmetical progression ; there-fore in the construction of the Table , there is no necessityfor calculating by the rule of proportion, nor even by plaindivision or multiplication, except in the first or uppermosttwo numbers in the first and second vertical columns ofanswers ; the difference of each of these numbers, so found,is the common difference of the progression in the firsthorizontal row of answers; then this common differencebeing added to that at the head of the second column givesthe next towards the right hand, or the uppermost answer atthe head of the third vertical column ; the common differ-ence being added to this sum, again gives the next succeedinganswer, and so on to the end of the horizontal progression :proceed with each vertical column by adding the commondifference in the same way throughout each progression, ifmore than one ; observing, that whatever number of timesthe common difference of any following progression in thegiven vertical column contains the common difference in thepreceding progression, so must the common difference of anyfollowing corresponding progression, in any vertical columnof answers, contain the common difference of the arithmeti-cal progression immediately preceding in the same column.
One thing may be observed in each vertical column, wherethe answers consist of integers only, and where there areseveral arithmetical progressions, from 1 to 10, from 10 to100, from 100 to 1000, and where each progression is formedof 10 terms, all the numbers of the first progression are tobe found respectively in as many of the first figures of thesecond progression, and all the figures of each number in thesecond progression are to be respectively found in as many ofthe first figures of each of the numbers in the third pro-gression, and so on, whatever be the number of progres-sions : therefore, if the last progression be constructed, thewhole of the progressions upwards may also be constructed :for whatever number of figures each number of the lowerprogression consists of, each number of the progression im-mediately above will respectively consist of the same, want-ing the right-hand figure ; likewise the third progression willhave the same number of figures in each number, wantingone, that each respective number has in the second progres-sion, or wanting two that each respective number in theundermost progression has. So that, by proper attention,one small Table, consisting only of 10 horizontal rows, willgive the answer for any number whatever.
In Tables where the given columns consist of differentdenominations, the answers may be first obtained in one de-nomination, and afterwards each of the answers may bedivided into the several denominations it will resolve into, inorder to facilitate the construction of the Tables.
i