(FORTUNE Magazine) – In medieval times monks cloistered in monasteries pondered ancient texts and scribbled with quill pens to find answers to such imponderables as how many angels can dance on the head of a pin. In the 1950s government-sponsored think tanks analyzed some of the cosmic questions of the day, like how many millions of fatalities would constitute an acceptable consequence of a nuclear war. Today a new class of well-compensated, highly educated specialists use computers and algorithms to quantify equally inscrutable issues, such as how valuable a crazy Chihuahua is to the Taco Bell restaurant chain.

Almost unnoticed, some half-dozen professional firms have sprung up in the past two decades with the mission of providing answers to similar kinds of questions. Clustered around university towns like Cambridge, Mass., and Palo Alto, these 21st-century think tanks are populated by economists, finance specialists, chemists, lawyers, and even some MDs. Most often their job is to put a dollar figure on actions, things, or ideas that haven't been measured before--the cost of a disability, the value of a stolen technology, the worth of a purloined trade secret. Their clients include corporations, governments, and trade associations, but their biggest customers are law firms. Now that lawsuits have replaced baseball as the national pastime, there is a seller's market for people who can attach pricetags to damages.

One fast-rising firm is the Analysis Group, headquartered on several expensively furnished floors of Boston's Prudential Center. Founded in 1981 by two alumni of now defunct Arthur D. Little, it employs some 250 professionals and generated revenues of $100 million last year, up 35% from 2001. The staff is about evenly divided between MBAs and Ph.D.s, and draws more brainpower from affiliated academics, including a former chairman of the Council of Economic Advisors and a Nobel Prize winner in economics.

Being a smart person for hire brings many kinds of work to your door, some of which the Analysis Group won't pick up. It doesn't take slip-and-fall or hot-coffee-in-the-lap cases. Nor will it get involved in such politically charged issues as tobacco liability suits. And it turns away contingency fees for fear that they might taint its independence. But those proscriptions haven't stopped it from taking on high-profile clients and assessing such issues as Satan's impact on Procter & Gamble's business and the value of the James Bond franchise to MGM. A noteworthy client these days is Microsoft, for which the Analysis Group evaluates potential damages in a host of antitrust cases.

Sometimes the firm's investigative work takes it into the field for research. Last year the Justice Department wanted to know whether the movie theater seats made available to handicapped moviegoers under the Americans With Disabilities Act were comparable to the seats offered ordinary patrons. Braving sticky floors and the smell of stale popcorn, Analysis Group researchers canvassed 140 theaters in dozens of multiplexes. They positioned themselves in the theaters 20 minutes before showtime to see which seats were occupied first, and then compared those with the seats offered to the handicapped. As yet there has been no determination in the case, though a number of theaters already have changed their seating policies.

At times the firm's research is wide of the mark. A decade ago Eli Lilly, the maker of Prozac, wanted to quantify the economic burden of depression. So the Analysis Group dispatched a six-person team headed by a health-care economist to figure it out. They calculated the direct doctor and hospital costs of depression and added the time lost by subpar job performance and absenteeism, plus the cost of depression-induced suicide. Using government studies on hospitalization, employment, salaries, and wages, they came up with a total of $43.7 billion a year.

Widely publicized, the number was picked up by Tipper Gore and used at a White House conference on depression. It turns out that it was way low. The team's leader now says it failed to consider the burden of depression on families, as well as the cost to individuals of diseases that accompany depression. He figures the true impact of depression is more than $100 billion.

And then there are the cases like the psychotic Chihuahua that are unexpected home runs. For more than a year Wrench, a two-man advertising agency in Grand Rapids, worked with Taco Bell to develop an animal mascot for its advertising campaign. A year later the Chihuahua turned up in the restaurant's advertising, but in a campaign created by another agency, TBWA\Chiat\Day. When Wrench didn't receive any licensing fees or royalties for the character, it sued, and its lawyers hired the Analysis Group to figure out how much the character was worth.

A team of eight analysts went to work to count Taco Bell's gains from the campaign. First they examined company documents quantifying the extent to which the dog increased advertising awareness. Then they developed econometric models to figure out how the increased awareness translated into food sales. That meant analyzing 12 years of weekly store sales data, adjusting for the impact of menu changes and other items, and then determining the effect on profits.

In the end the analysts found that the Chihuahua had an incremental impact on sales of 1% to 4%. The case wound its way through several courts over five years before a U.S. district court ruled in June in Wrench's favor and awarded the two men damages of $30.1 million. In their own quiet way, members of the Analysis Group celebrated the victory. Said one principal: "That little dog captured the imagination of America, and we captured the value of Wrench's imagination in creating it."