Field: Plot =-. P and F being always reduced to the same denomination.

Thus a scale of fjvis 's 5280 ft. of field to 1' of map or one mile to 1 ft. — .fa of a mile to 1'', and not 12" to 1 mile

It is evidently incorrect therefore to indicate the scales of maps as so many inches to a mile as is frequently done. Take the case of the recent Geological maps of one of our sister states said to be plotted on a scale of 3" to 1 m- or 3" to 63,360" = «-3-|«° = 21,120 that is to say the map is 21,120 times larger than the state itself, a manifest absurdity resulting from considering the map as the first quantity or standard rather than the field itself.

In such cases errors of interpretation can scarcely arise as the intention is so evident, but there are numerous others that may lead to misconstruction, as where the drawings of small objects are nearly of the same size as the things represented—thus a scale of £" to 1" would confuse a mechanic unless he happened to know which was the larger, the object or the drawing.

So the expression \" to 1' is likewise incorrect as it is the reciprocal of the ratio intended—the inches evidently referring to the drawing and the foot to the object. As it stands, applying the definition of ratio as deduced, it will be equal to 12 -s- \ = 48, making the drawing 48 times the size of the model—it should be 1' to \".

If it be remembered that the antecedent always refers to the given object and the consequent to the drawing, no difficulty can arise. It will always happen then that if the drawing is on a smaller scale than the thing delineated, the ratio will be a proper fraction; if larger, an improper fraction, and if equal the value will be unity, or {.

It is hardly necessary to call attention to the fact that the number of scales in use is practically infinite, and that serious inconvenience results therefrom to Engineers and Surveyors whose work extends over several counties or states, making it frequently necessary to re-draw large sections of country. In compiling atlases it is the practice of publishers to vary the scales according to the amount of territory to be represented that the sheet may be filled up, but nothing is gained thereby since the scale used for the greatest area to be represented will show with equal clearness all the features of any other area. Moreover the eye becomes accustomed to estimating distances on the maps, with sufficient accuracy for a reconnaissance, when the scale is uniform, but when variable it leads to great confusion, and especially when the publisher has neglected to indicate the scale, as sometimes happens.

It is very desirable to establish, if possible either by recommendations of scientific societies or by general laws, some conventional scales for maps of various sizes. Taking a state of medium area as N. Y. or Penna. for the unit, and reducing it to a convenient size sheet of paper, say 4 X 3 ft., would require a scale of i^Vinr- tne same as is used by the U. S. Coast Survey for general charts and reconnaisance, but too small for most other purposes. Larger states could be plotted on the same scale by dissecting them. Foreign countries conducting Geodetic Surveys have adopted such a system. In Prussia, Austria and Switzerland the plane table sheet are plotted on a scale of yjfojj. In Italy the field work is plotted on a scale of yffJinr, and in Sweden xWtsins- Tne older British charts and maps were made on a scale of 1 m- to 1" or sjfair. and the later maps of 1 "'■ to 6" or TTT5BTi> DUt these latter, while not being large enough to show parish boundaries with sufficient accuracy, require about six times the amount of labor in their preparation and are inconvenient. The scale used by Prussia and Switzerland for general maps is xMrtatr> or one fourth that of the detail sheets obtained from the plane table surveys.

Populous, cultivated and mineral districts in Great Britain are plotted on a scale of 5Sy„ — 1 m- to 25.344", partially cultivated and thinly settled districts, on a scale of 1m- to 6" = Yshn- For tne plans of cities of over 4000 inhabitants a scale of -J5 or 1 "■ to 10.50 feet is used, and for towns and villages r^ s or 1 m- to 5 ft. is general.

Numerous other instances might be cited showing the great variety of scales in use, but these will suffice. It is evident that in Government or State Surveys some systematic connection may readily be established between the"several scales used, and it is very desirable that this uniformity of scale be made more general. The scale adopted should be just large enough to show clearly all necessary detail. Anything more than this is a wasteful expenditure of time and money.

For general maps of States showing intercommunications, a scale of TTrtWr will be found sufficiently large.

For maps of counties, in toto, a scale of -gvhsis w'" suable all necessary features to be clearly represented; this scale applied to Lycoming Co., the largest in Penna.. would require a map 6J X 4J, ft. For townships the scale of 23550 is quite large enough, and furnishes an admirable size for the projection of Geological data.

For cities, towns and villages some decimal, sub-multiples of the above scales should be used.

Cadastral maps of farms, parks or estates may be plotted on scales of

1 1 l ptr>

In indicating the degrees of slopes or the bater of retaining walls, the natural tangent of the angle which the slope makes with the horizon should invariably be used.

To save time in determining the relative values of some of the most important scales in use, and to aid in introducing the metric system of lengths, I have with the assistance of Messrs. Win. M. Potts and J. W. Van Osten, Jr., prepared the accompanying tables of equivalents. The first, gives the number of Miles, Kilometers, Poles, Chains, Yards, Meters and Feet of territory which are equivalent to one inch of map for any given scale. The second, is the reciprocal of the first, and states the amount of map surface which would be covered by any one or more of the above units, for any scale.

FROC. AMElt. PHILOS. SOO. XVIII. 102. G. PRINTED DBC. 20, 1878.