WRITING THE NEXT federal budget, as the administration is doing it, comes down to one crucial question: how much for the Defense Department? Everything else depends on the answer to that one. President Reagan has started off with an adamant refusal to entertain any thought of tax increases. The result is that all progress toward a lower deficit has to be made by cuts in spending alone.

This budget, for the fiscal year beginning next October, would run to about a trillion dollars under present spending policies. To get the deficit down to its target of 4 percent of the gross national product, the administration will need to cut that amount by at least $40 billion. At first glance, cutting $40 billion out of a trillion-dollar budget doesn't look terribly difficult. But the three largest categories of outlays are already immune to cuts -- defense and Social Security by presidential decree, and the interest on the national debt by necessity. Together they comprise two-thirds of the budget. The $40 billion has to come out of the remaining third. Now the puzzle gets a bit more challenging.

That vulnerable third of the budget includes everything from medical research to farm subsidies to environmental protection. And perhaps much more than $40 billion will have to be taken out of it. If other programs -- i.e., defense -- rise faster than revenues do, the $40 billion will have to be raised to offset those increases. But defense spending is currently rising much faster than revenues. If it goes as high in the 1986 budget as the estimate that the administration published last summer, that would require an additional cut of $10 billion in the vulnerable third of the budget.

That's why the president has to settle the quarrel over the Defense Department's share of the budget before his staff knows how much it has to eliminate elsewhere. At $50 billion, the cuts would be nearly one dollar out of every six in the vulnerable third, a proportion that strains the limits of the possible -- particularly since the administration's target also requires similar reductions in each of the following two years.

Above all, the cuts designated in this budget have to be believable. Proposals to sell the Washington Monument or to auction off the space shuttle won't do. The administration argues that it was reelected to hold taxes down, to get defense strength up and to diminish the deficit. The budget struggles now going on at the White House are testing whether the president is, in reality, able to fit those three commitments together. The chances will be very low if defense spending continues on its present track.