by Leonardo da Vinci
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But if you do not wish to strengthen the arch with an iron tie you must give it such abutments as can resist the thrust; and you can do this thus: fill up the spandrels m n with stones, and direct the lines of the joints between them to the centre of the circle of the arch, and the reason why this makes the arch durable is this. We know very well that if the arch is loaded with an excess of weight above its quarter as a b, the wall f g will be thrust outwards because the arch would yield in that direction; if the other quarter b c were loaded, the wall f g would be thrust inwards, if it were not for the line of stones x y which resists this.
787.
Here it is shown how the arches made in the side of the octagon thrust the piers of the angles outwards, as is shown by the line h c and by the line t d which thrust out the pier m; that is they tend to force it away from the centre of such an octagon.
788.
An Experiment to show that a weight placed on an arch does not discharge itself entirely on its columns; on the contrary the greater the weight placed on the arches, the less the arch transmits the weight to the columns. The experiment is the following. Let a man be placed on a steel yard in the middle of the shaft of a well, then let him spread out his hands and feet between the walls of the well, and you will see him weigh much less on the steel yard; give him a weight on the shoulders, you will see by experiment, that the greater the weight you give him the greater effort he will make in spreading his arms and legs, and in pressing against the wall and the less weight will be thrown on the steel yard.
789.
The first and most important thing is stability.
As to the foundations of the component parts of temples and other public buildings, the depths of the foundations must bear the same proportions to each other as the weight of material which is to be placed upon them.
Every part of the depth of earth in a given space is composed of layers, and each layer is composed of heavier or lighter materials, the lowest being the heaviest. And this can be proved, because these layers have been formed by the sediment from water carried down to the sea, by the current of rivers which flow into it. The heaviest part of this sediment was that which was first thrown down, and so on by degrees; and this is the action of water when it becomes stagnant, having first brought down the mud whence it first flowed. And such layers of soil are seen in the banks of rivers, where their constant flow has cut through them and divided one slope from the other to a great depth; where in gravelly strata the waters have run off, the materials have, in consequence, dried and been converted into hard stone, and this happened most in what was the finest mud; whence we conclude that every portion of the surface of the earth was once at the centre of the earth, and _vice_versa_ &c.
790.
The heaviest part of the foundations of buildings settles most, and leaves the lighter part above it separated from it.
And the soil which is most pressed, if it be porous yields most.
You should always make the foundations project equally beyond the weight of the walls and piers, as shown at m a b. If you do as many do, that is to say if you make a foundation of equal width from the bottom up to the surface of the ground, and charge it above with unequal weights, as shown at b e and at e o, at the part of the foundation at b e, the pier of the angle will weigh most and thrust its foundation downwards, which the wall at e o will not do; since it does not cover the whole of its foundation, and therefore thrusts less heavily and settles less. Hence, the pier b e in settling cracks and parts from the wall e o. This may be seen in most buildings which are cracked round the piers.
791.
The window a is well placed under the window c, and the window b is badly placed under the pier d, because this latter is without support and foundation; mind therefore never to make a break under the piers between the windows.
792.
A pillar of which the thickness is increased will gain more than its due strength, in direct proportion to what its loses in relative height.
