420

HISTORY OE ENGINEERING.

Book I.

move the nails from o to K, according to themargin you intend to show in the front, and.from them let fall the arch m , and that will limitthe border or margin, which may be divided, anda mould made to fit the curve in the same manneras if the arch was in any other form, as you maysee in the figure, which in practice must be turnedupside down; but, as I observed before, that thelower the line was let fall, the nearer it approachedan hyperbola, therefore it cannot be expected thatthe lines to and m will be parallel or appear con-centric like arches struck from the same centre,but those that should think it disagreeable for the margin of an arch to be wider at top,may easily prevent it by making the moulds parallel to the inward arch, which would be amatter of indifference in respect to its strength.

“ But, notwithstanding all those distant appearances before-mentioned, that arc made bythe catenarian line, yet I cannot help thinking that they are all of the hyperbola kind, foras the hyperbola is that section of the cone cut any where when parallel to its axis, I say,that it is no hard matter to imagine a cone large enough that will admit of the flattest aswell as the highest arches to be cut off by such a section ; now, I believe it will appearfrom what I have already said, that catenarian arches are the best shape for duration, andfor that reason the most proper form for the new bridge on the river Thames , if it be builtwith stone.

“ A cask or tub may very justly be said to have the same properties as an arch, for thestaves may be considered as so many stones, whose joints point to one common centre, andthe hoops that circumscribe them as their weight or pressure; then I think it cannot bedenied that the tighter these hoops are drove on round such a vessel, the more compact itmust keep the staves, and the stronger the vessel will be for such a weight or pressure;therefore it appears to me that weight on an arch will have the same effect as a hoop hathon a tub, if the weight be well proportioned.

“ But in order to give my opinion in this affair, the actions of gravity on solid bodies oughtto be understood, and the shape of the arch well considered before a judgment can be formedabout the weight it will carry, or on what part to bestow it to the best advantage; thereforeI shall lay down the following theorems: —

“ I. When any weight presses on the sides or haunches of an arch, the weight on thecrown must be considered as an abutment to that weight on the haunches, and in proportionto the height they rise from the chord lines of their respective segments will the weight onthe crown be required.

“ II. And as the weight on the crown is an abutment to the weight on the haunches, so isthe weight on the haunches an abutment to the weight, on the crown.

“III. And as the weight on the haunches wants to reduce the curvature thereof to astraight line, contrary to that doth the weight on the crown want to make it rise higher orpush it out at foot, if there is no abutment placed there to prevent it, and both wouldproduce that effect if it was not for a counterbalance of power, as will be best explained bythe following figure.

“ If the segment aec will bear five hundred weight, the semicircle a be will bear six withthe same abutment, and the cate-narian a d c will bear the weight inrespect to the abutments below, butif they are both considered as havingno weight on their haunches, thecatenarian must be the strongest byreason that the curvature of itshaunches are nearer a right linethan those of the semicircle, for wasthere as much weight laid on thecrown of the catenarian arch as itwould just bear, the same weight ifapplied to the semicircle wouldoccasion it to burst out and falldown, which plainly sheweth thatthe semicircle wants a greater weight on its haunches to counterbalance the same weight onthe crown; and how much more weight is required on the haunches of the semicircle tocounterbalance the same weights on the crowns of both may be nearly discovered by thesame method, for drawing the chord line o, will cut off one haunch of both, which may beconsidered as whole arches, and the difference of weight be discovered in the same manner,for if the haunch of the catenarian requires nine hundred weight, the semicircle will require

b

Fig. 424.

Fig. 423.