| In 1900, Davidson had been
admitted to the bar in Washington, D.C., and Kentucky, and in 1903, he was
allowed to practice before the D.C. Court of Appeals. In 1912, Davidson
was admitted to the bar of the U.S. Supreme Court, when Belva A. Lockwood,
a progressive white woman lawyer, nominated him.
In his capacity as clerk in the U.S. Department of the Treasury, Davidson patented at least two devices that he thought would make work there more efficient and economical. Both involved the adding machine. One of his inventions was an attachment to adding machines that rewound the roll of paper so that the back of the paper could be re-used (1908 patent and drawing).
He also developed an adding machine attachment that would automatically calculate the totals for money orders and fees. According to Davidson, the purpose of this second invention was to
"... provide means, applicable to adding machines of various types, whereby when the machine is operated in the usual way to record the amounts of checks, drafts, orders, bills, or any principal sums, it will automatically record, in a separate column, different amounts representing fees, penalties, charges, discounts, etc., as desired, according to a predetermined scale bearing some arbitrary fixed relation to the principal amounts.
In the United States postal money order system, for instance, money orders are sold for amounts ranging from one cent to $100.00, inclusive, and fees are charged for the issuance of the orders varying from three to thirty cents, according to the amount of the order.... When these money orders have been cashed and returned to the auditing department, adding machines are used to add and total the amounts of the orders. When a machine is operated to record and amount of a money order, the operator touches a key or keys in another part of the machine to record the fee charged for the amount of that particular order. This entails touching two sets of keys for each money order, and the operator, although familiar with the scale of fees, is liable to make a mistake and record the wrong fee and thus cause discrepancy in the results.