The demonstration of a plain scale, by which you may work several lines in proportion one to the other

It is necessary before we proceed any further to make a scale to enable us the better to lay down our geometrical problems. Besides, it will perfect the young learner in its use for shipping, without which he cannot well proceed. And to make it more easy perceived, it is a piece of board, brass or paper, whereon you make with your compasses several equal parts, greater or lesser as your plate or design will afford, the larger the better regarding the design in hand, and considering a ship, a body of many feet, which will take up a great room.

I choose to suppose a quarter of an inch to a foot, or as my plate will admit of, more or less. This being done, I strike two straight lines of equal length, which I divide into 24 equal parts, beginning at 1,2,3,4,5,6, until I have procured 24. When it is thus divided, subdivide every part into eight parts to make it into half quarters of each part if your scale will admit thereof. Having now made your scale, it is applicable to either miles, roods, yards, feet, or money, as pounds, shillings, pence, labour, and the like.

Therefore it is of absolute necessity to be very exact in the making of it, by reason of error which will other ways attend your work. Being thus prepared by any scale of equal parts, I shall proceed in the next place unto its use, this being only to prescribe the scale and its shape, which was required to be demonstrated.

Note that a scale is only equal parts, and it in proportion to whatever you apply it. If you apply it to inches it bears proportion as 1 to 4. This scale being t part of an inch, if to feet it bears as 1 to 48. If to yards, it is as 1 to 144, and so accordingly to whatever you apply it, as you shall find hereafter.

Having two -lines given, to find a third proportional line to them. Your scale now having been made as on the other side, I shall proceed to the drawing of lines in proportion the one to the other, as for example, the line A is eight feet, and the line B is 12 feet. I desire a third line in proportion to A, as A is to B. You must work thus. Strike at adventure 1 two lines meeting at one end as an angle.

Let the two lines be LMN. Take from your scale the line A, which is eight feet, and set one leg at M, the other at D. Having so done, take the line B, which is 12 feet, and place it from M to E, and then draw the line D and E. Having so done, take again the line B which was 12 feet, and set it from M to H. Lastly take the distance from H to D and strike a line parallel to ED, which will be the line HI, 18 feet, and is in proportion from I to M as the line B is to A, as appears by the figures demonstrated in the margin.

By this method are all lines drawn by geometry, so that it is needful for the young artist to bear this work in remembrance, and so I shall say no more of this nature, but pass to the rest of the work, in which I will be very brief, only according to my promise.

Having three lines given, to find a fourth proportional line to them. You must proceed in this work as in the former, striking two lines at adventure, like to an angle meeting at each end, supposing them to be the lines KGD, meeting at G. 2 Having so done, the question which is proposed is to have three lines given to find a fourth, as the lines ABC, the line A being 18 feet, the line B 14, the line C 10. I would have a fourth line in proportion to A as B is to C.

Work thus. Take with your compasses the line C which is 10, and set one leg of your compasses at G, setting the other leg at C. Having so done, take the line B, which is 14 feet, setting one leg at G, the other at B, and then strike the line BC. Having so done, take the third line A, which is 18, and set one leg at G and the other at A.

Then strike a parallel line to BC, which is the line AF, and then is the line from F to G 251o, which is the fourth line sought for, being in proportion to A as B is to C. And in like manner is all work of this nature concluded, which I pray remember before you pass into the next.