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to exhibit all the varieties of each case, was what appeared to the writer to be chiefly wanting, and which gave rise to the following compendium.

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The Problems are, it is presumed, arranged in that order which is best calculated to promote progressive improvement, and the language in which the rules are delivered, it is believed, will not be found destitute of perspicuity and accuracy: all obscure or scientific terms which were not absolutely necessary have been studiously avoided; and the utmost endeavours to simplify and adapt the Work to the capacities of youth have been exerted.

As this Work was intended to furnish a practical system only, all such positions as required diagrams or other mathematical illustration have been omitted; the young student being referred for full and philosophical rationale and demonstration to works on astronomy, spheric trigonometry, and

the projections of the sphere, many of which are extant, by authors so well known to the public, as to render the mention of their names superfluous.

With respect to the tabular part, several new and useful arrangements will be found, most of which have an immediate and particular reference to the contents of the Work; a circumstance which will render this Treatise more complete than most others of the same price and dimensions.

Upon the whole, no exertions have been wanting to adapt this Work to the purposes of scholastic instruction and to private students, by the numerous and extensive variety of examples to each problem. And that this branch of education might be still more facilitated, and the difficulties and labour both to teachers and scholars in a great degree removed, an "EXAMPLE BOOK," adapted to the Work, by the same Author, is nearly ready for publication, in which

the problems and questions in the "PRACTICAL TREATISE" have been arranged with proper spaces and columns for the insertion of the answers at large.

This "EXAMPLE BOOK" is prepared in the usual manner of a ciphering book, on suitable paper for writing, and may be had with or without the primary Work.

MARLBOROUGH HOUSE,
Walworth, July 25, 1810.

Prob. XV-A place and hour, being given, to find

what places on the globe have any other hour proposed 25

Prob. XVI-A place on the globe within the tropics

being given, to find the months and days of the year

on which the sun will be vertical to the same....

Prob. XVII-The month and day being given, to find

all those places to which the sun will be vertical.. 27

Prob. XVIII-The month and day being given, to find

that place in the torrid zone to which the sun will

be vertical at a given hour in a given pluce.....

Prob. XIX-The day and hour at any place being

given, to find all those places where the sun is then

rising, where it is noon, where the sun is setting,

where it is midnight, and where it is day and night 28

Prob. XX-Place, day, and hour being given, to find

the length of the day or night on any part of the

globe between the equator and the polar circles....

Prob. XXI-A place and the length of the day being

given, to find the day, the sun's place, and declination 31

Prob. XXII-The latitude and day being given, to

find how much the sun's declination must vary to

make the day an hour longer or shorter than the

given day....

Prob. XXIII-The length of the longest day, not ex-

ceeding 24 hours, being given, to find the latitude of

the place..

Prob. XXIV-The longest day in any latitude being

given, to tell in what other latitude the longest day

is 1, 2, 3, &c. hours longer than in the given place

Prob. XXV-To find the length of the longest day

or night in either of the frigid zones.

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