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129)

[[left margin]] Talbot's Micrometer p.110 considered. [[/left margin]]

This last observation is Geometrically true, but so insensible that it cannot be regarded in practice; for the tangents and arches of the first 30 minutes, to a radius of about 9 Feet, do [[strikethrough]] so [[/strikethrough]] ^ [[insertion]] very [[insertion]] nearly coincide & increase together, as will appear from the following Table. To calculate which say 64 Threads of his screw : 1 Inch :: 1 Thread or 1/2 turn of the Screw :,015625 inch, 30 of which =,46875, which subtends an angle of 15' to the Radius 10'1,429 inches (found by his analogy on p.110) the focal length of his object Glass; to which radius the quantity of every thread or 1/2 turn of the Screw I have calculated, as under, with their tangents to Radius of Tables.
4. It appears to me that this micrometer may be adapted to all Radi [[insertion]] i [[/insertion]] [[strikethrough]] us [[/strikethrough]] & Screws. 1 [[superscript]] st [[/superscript]] Suppose only a pin at e fig.1 2 then, had his Screw been only 32 threads to an inch, the wheel f might have consisted of 30 teeth as before ^ [[insertion]] or of any [[strikethrough]] number of [[/strikethrough]] other number; [[/insertion]] but the index i then describing 2 minutes in every revolution, the plate, upon which the index a shew the seconds, must be divided into 120 equal parts. [[strikethrough]] or the wheel f might have consisted of 15 teeth, & the plate shewing seconds; divided into 60 parts, because then one revolution of i would be a minute. [[/strikethrough]] Again, Suppose [[strikethrough]] the [[/strikethrough]] [[insertion]] his [[/insertion]] Screw of 64/5 = 12,8 threads to an inch, the wheel f might [[strikethrough]] be [[/strikethrough]] contain [[strikethrough]] 30/6 & [[/strikethrough]] 6 ^ [[insertion]] or any other number of [[/insertion]] teeth, for i to shew a minute in a revolution, & the plate at a divided into 60 6 [[insertion]] x5=300 [[/insertion]] parts for seconds: Also, had his Screw contained 128 threads in an inch, the wheel f might contain any number of teeth, but the plate at a divided only into 30 equal parts for seconds, because the index i turns twice round in a minute.
2. Suppose now a wheel at e instead of a pin only, then may Minutes and second be shewn [[strikethrough]] only [[/strikethrough]] by inspection only with a Screw ab containing any given number of threads in an inch. Thus, Adapt the two wheels at f and e so as the index i may, in one revolution, describe one minute which striking another wheel of 30, [[strikethrough]] teeth [[/strikethrough]] 31, or 32 teeth to count its number of Revolutions or Minutes, and the ^ [[insertion]] n [[/insertion]] itself shewing seconds upon the plate AB (under it) divided into 60 equal parts.

[[table to left of previous paragraphs]]
Turns or Revolutions of the Screw | Tangents to Rad. of Tables, [[strikethrough]] of [[/strikethrough]] at each turn of the screw, from the center. | Quantities of those tang.[[superscript]] ts [[/superscript]] described by one hair from the center. 

1 | ,000145445 | 0' 30"
2 | ,000290890 | 1" ·
3 | ,000436335 | 1" 30
4 | ,000581780 | 2" ·
5 | ,000727225 | 2" 30
6 | ,000872670 | 3" ·
7 | ,001018115 | 3" 30
8 | ,001163560 | 4" ·
9 | ,001309005 | 4" 30
10 | ,001454450 | 5" ·
11 | ,001599895 | 5" 30
12 | ,001745340 | 6" ·
13 | ,001890785 | 6" 30
14 | ,002036230 | 7" ·
15 | ,002181675 | 7" 30
16 | ,002327120 | 8" ·
17 | ,002472565 | 8" 30
18 | ,002618010 | 9" ·
19 | ,002763455 | 9" 30
20 | ,002908900 | 10" ·
21 | ,003054345 | 10" 30
22 | ,003199790 | 11" ·
23 | ,003345235 | 11" 30
24 | ,003490680 | 12" ·
25 | ,003636125 | 12" 30
26 | ,003781570 | 13" ·
27 | ,003927015 | 13" 30
28 | ,004072460 | 14" ·
29 | ,004217905 | 14" 30
30 | ,004363350 | 15" ·
[[/table]]

Examples. 1. Suppose I have a telescope, whose object Glass hath a focus of 2 feet; this by table on p. 128 requires a screw of 287 ^[[insertion]] threads [[/insertion]] [[strikethrough]] teeth [[/strikethrough]] in an inch, which is impract [[strikethrough]] ally [[/strikethrough]] ^ [[insertion]] icably [/insertion]], & my Screw hath only 41 ^ [[insertion]] threads [[/insertion]] [[strikethrough]] teeth [[/strikethrough]] in an inch. which is exactly 7 times to few; therefore the index i must make 7 revolutions whilst the Screw ab makes one: whence if the wheel at e contains ^ [[insertion]] 42[[insertion]] [[strikethrough]] 41 [[/strikethrough]] teeth, and that working in it at f [[strikethrough]] 4 [[/strikethrough]] 6 teeth, then i goes one round in a minute, driving another wheel of any convenient N. [[superscript]] o [[/superscript]] of teeth so as to tell or count the revolutions of i; which i also shews the seconds upon the plate under it divided into 60 equal parts. 
Example. 2. Suppose the focal length of the object Glass 20 Feet. & my screw 58 threads in an inch: I find by calculation that one turn of the screw is only half a minute; therefore ab must go twice round to a minute. From whence the wheel f must [[strikethrough]] be [[/strikethrough]] ^ [[insertion]] have [[/insertion]] 12 teeth, double that of e, of 6 teeth only: rest as before, in the last example. -- N. B. the index i may turn twice round in a minute, if it should be required in any case, and the wheel which it turns must then have 60, 62 or 64 teeth, number every other for minutes, & those between will be 1/2 min. [[superscript]] s [[/superscript]] the seconds of which 1/2 minutes will be shewn upon the plate under i. divided into 30 pts. 

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