Babbage's Difference Engine
手順
Work the method of differences by hand
Work the method of differences by hand
Do it on paper first. The trick is so simple it looks like a trap.
- Take the polynomial f(x) = x² + 2x + 3 and tabulate it for x = 0,1,2,3,4,5.
- Write the differences between consecutive values underneath — that is the first difference row.
- Write the differences of THOSE — the second difference row.
- Look at the second difference row.
The second differences are all the same number. For any polynomial of degree n, the nth difference row is constant — and that means you can run the table BACKWARDS: start from the constant, add upward, and generate the next value using only addition.
Extend your table two more rows without ever squaring anything. Add the constant to the last second difference, add that to the last first difference, add that to the last value. Three additions produce the next point of a quadratic.
This is why the machine has one column per difference order and no multiplier anywhere. Babbage did not build a machine that evaluates polynomials; he found a way to evaluate polynomials that needs only a machine that adds.このステップの材料:
Graph Paper1 pad必要な工具:
CalculatorRun the engine in code and watch error propagate
Run the engine in code and watch error propagate
必要な工具:
Desktop ComputerOpen the mechanism in 3D
Open the mechanism in 3D
Rotate the model. Four vertical columns, six digit wheels each — one column per difference order, one wheel per decimal digit, and every wheel has ten teeth because it counts 0 to 9.
Between the columns run the carry arms. Those are the hard part of the whole machine: when a wheel passes 9 it must nudge its neighbour, and if several wheels are at 9 the carry must ripple all the way along. Babbage’s anticipating carriage was designed to handle a whole rippling carry in one motion instead of one wheel at a time, and it is the single most intricate part of the design.
Open it in Blender and count the teeth. Then consider the manufacturing problem: the full engine needs about 25,000 parts, and every one must be accurate enough that no wheel ever slips a single tooth — because the notebook just showed what one slipped tooth does.必要な工具:
Desktop Computer
Digital Caliper 6-InchCut a digit wheel and a carry cam
Cut a digit wheel and a carry cam
Make the two parts the machine is built from, and feel the precision problem.
- Cut a ten-tooth wheel in brass — index the blank at 36° intervals and cut each tooth.
- Measure the error in each tooth position with a caliper against a dividing plate.
- Make a second wheel with a single carry cam projecting from one face, positioned to act as that wheel passes from 9 to 0.
- Mesh them and turn slowly, watching the carry transfer.
The carry cam must act ONLY between 9 and 0 and never at any other position, which means its angular width and timing are as critical as the tooth spacing.
Reverse-engineering note: Babbage's engines were never completed in his lifetime, and the standard story is that Victorian machining was not accurate enough. That story is wrong. The Science Museum built Difference Engine No. 2 in 1991 from his drawings, to tolerances available in the 1840s, and it worked — 8,000 parts, 5 tonnes, and it computes correctly. The obstacle was cost, management and Babbage's habit of redesigning, not precision.このステップの材料:
Brass Round Bar (25mm)1 個
Brass Sheet (0.5mm)1 枚必要な工具:
Metal Lathe (Benchtop, 7x14)
Milling Vise (4-inch)Digital Caliper 6-Inch
Micrometer
File Set
Clear Safety GlassesPrint the answer, because the human is the weak link
Print the answer, because the human is the weak link
Trace both paths. The engine's purpose was never to display a number — it was to produce a printing plate, and Babbage understood that a perfect calculation transcribed by a tired clerk is not a perfect table.
That is why roughly half the design is a printing and stereotyping apparatus, and why the machine is enormous. He was not automating arithmetic; he was automating the entire pipeline from computation to printed page, because every human step in that pipeline was a place errors entered.
The principle generalises well beyond tables: a system is only as accurate as its least reliable transcription step, and adding a perfect component in the middle of a chain of sloppy ones buys nothing. It is the same reasoning that made Haldane publish TABLES rather than a physiological model, in the sub-sea batch — the deliverable has to reach the person doing the work in a form they cannot corrupt.Flow
必要な工具:
Desktop Computer材料
3- 1 padプレースホルダー
- プレースホルダー
- プレースホルダー
必要な工具
8- プレースホルダー
- プレースホルダー
- プレースホルダー
- プレースホルダー
- プレースホルダー
- プレースホルダー
- プレースホルダー
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