Understanding what was going on with the doubled lenses was no easy thing. For Galileo, everything physical came down to matters of geometry, and clearly this bending of the light was a geometrical action; but he lacked any laws of refraction, and could not discover them merely by substituting lenses one after the next. There were tangible variables involved, however, that they could subject to the workshop techniques they had already honed in previous pursuits.
So the workshop’s artisans met in the hour after sunrise, some of them servants of the house, others local ancients retired from arsenals, or lads from the neighbourhood; still rubbing the sleep from their eyes, squeezing the bellows to get the fires in the furnaces going, picking up the work they had laid down the night before. They followed Galileo’s routines: they measured things twice, wrote everything down. They worked while breaking their fast. Watched the rainstorms out of the open side of the shed, waiting for the light to get better so they could get back to work. The brick furnace was a bulwark just outside the roof, and they could stand near the back of it and stay warm while the rain came down, although as it was summer the afternoon thunderstorms weren’t so cold. The large central area of the shop was earthen-floored and held several long tables, one of them under the back wall devoted to all their tools. In the dim rain-light they could clean or sharpen tools, put things in order, pick away at the goose carcass from the night before. When the sun came out they returned to the work.
They made so many alterations in every new spyglass, Galileo was not quite sure what change was having what effect; but it was too interesting to slow down and isolate the variables to make sure of things, except when pursuing a crucial point. The epistemology of the hunt was to follow one thing after another, without much of an overall plan. They found that tubes made of cardboard, sometimes reinforced by slats or covered with leather, worked perfectly well; the interiors did not have to be perfectly smooth, although one saw a clearer image if they were painted black. Most important were the lenses. The one next to the eye they called the eyepiece, the one at the far end, the objective. Both concave and convex lens surfaces, if properly ground, constituted sections of spheres, bulging either in or out. Spheres of differing radii gave different curvatures. The radius of the complete sphere that was implied by a lens, Galileo called its focal length, following the lensmakers’ usage. Fairly soon their repeated trials with different lenses revealed that larger magnifications resulted from a long focal length for the convex lens at the far end of the tube, combined with a short focal length for the concave lens of the eyepiece. Grinding the convex lenses was easy enough, although it was important to eliminate small irregularities if possible, as these made for blurred patches. Grinding truly smooth curved depressions into the much smaller concave lenses, however, was harder to do. A small ball set in a rotating steel-milling mechanism that they screwed to one of the work tables served as their grinding tool. To see better they wore spectacles made of lenses ground earlier in the effort.
While this was going on Mazzoleni was also making cardboard tubes that would snug into his main tubes of leather and staves, giving them the ability to adjust the distance between the lenses and thus sharpen the image. The eyepieces were smaller, so they put the drawtube at that end, and fitted it with felt shims.
To find out what degrees of magnification they were getting, Galileo affixed a gridwork to a whitewashed part of the garden wall. This enabled him to measure accurately the difference between the enlarged image of the grid and the image he saw through the other eye at the same time.
On the afternoon of the seventeenth of August, Galileo examined their three best performers. All were about the same length, which was just over a braccio, as measured by their in-house yardstick. Studying the notes, Galileo compared all their dimensions, scribbling more notes as he did so.
All at once he laughed out loud. One of his special moments had come again, a flash of sudden insight at the end of a period of investigation, giving him a jolt and a shiver, as if he were a bell and the clapper had just tapped him. He shouted, ‘MAT ZO LEN EEEEEEE!’
The old man appeared, more dishevelled and whiskery than ever, red-eyed with lack of sleep. ‘Look!’ Galileo commanded. ‘You take the focal length of the objective-for this one, a hundred minims-and you divide that by the focal length of the eyepiece-in this case eleven minims-and you get a number which identifies the power of magnification of the device, thus here about nine times! It’s a ratio! It’s a ratio, it’s geometry again-’ He seized the old man by the shoulder: ‘Not only that, but look! Subtract the eyepiece focal length from the objective focal length, and you get the distance apart that the lenses are when the thing is focused properly! In this case, just short of one braccio. It’s a simple piece of subtraction!’
At this realization he grew somewhat glorious, as he often did when he was able to say new things of that sort. He congratulated everyone in the household, called for wine, threw crazia and other small coins at the servants and students who poured out into the courtyard to join the celebration, hugged them one by one while he was giving thanks to God and also indulging his most boastful humour, which was something to witness. He praised his genius for coming through for him again, he danced, he laughed, he grabbed Mazzoleni by the ears and shouted in his face:
‘I’m the smartest man in the world!’
‘Probably so, maestro.’
‘The smartest man in history!’
‘That’s how much trouble we’re in, maestro.’
This kind of poke in these moments of glory would only make him laugh and toss Mazzoleni aside, to be able to continue his jig. ‘Florins and ducats, crowns and scudi, I’ll buy Rachel and I’ll buy Trudi!’
No one in the household understood quite why he believed the glass was going to make him rich. The servant girls thought he meant to use it to watch them doing the laundry down at the river, which he did already from what he thought was a discreet distance.
Eventually everyone went back to work. Mazzoleni was left holding the glass, shaking his head at it. ‘Why should there be such proportions?’ he asked.
‘Don’t ask why.’ Galileo snatched up the glass. ‘Why is what our philosophers ask, and that’s why they’re so full of shit. Because we don’t know why. Only God knows why. If He does.’
‘All right, I know. Just ask what, just ask how. Still. You can’t help but wonder, can you.’ Waving at the new page of Galileo’s folio, filled with diagrams and numbers. ‘It seems so…’
‘So neat? Yes. Quite a coincidence, for sure. Quite the what-have-you. But it’s just more proof of what we already knew. God is a mathematician.’
As a mathematician himself, Galileo found saying this sentence immensely satisfying; often it was enough to bring tears to his eyes. God is a mathematician. He would emphasize the thought by taking a hammer to their anvil. And indeed the thought rang him like a bell. He would bring his hands together as if in prayer, and take a deep breath and expel it tremulously. To read God like a book; to solve him like an equation; it was the best sort of prayer. Ever since that time when he was a boy, he would explain, when he had looked up in church and seen a lamp swinging on its chain, and realized by timing it to his pulse that it took the same time to make its sweep back and forth no matter how far it was swinging, he had felt the direct touch of God in all these things. There was a method to His madness, clearly, and that method was mathematics. This was a comfort when the madness seemed all, as when he was sick, or in pain, or struck down by melancholy; or witnessing the effects of the plague; or contemplating the immense realm of human wickedness. Then his only comfort was the world’s inherent geom-etries.
The day for his Venetian demonstration approached, and their best tube showed