Bring 8 cups of water to a boil over high heat in a medium saucepan. Meanwhile, using a mandoline fitted with the julienne blade, slice celeriac into julienne matchsticks. Quickly cook celeriac in boiling water until slightly limp but still crunchy, 45 to 60 seconds. Transfer immediately to ice water. When cool, drain well, pressing any excess water through a sieve. Transfer to a medium bowl.
Adjust mandoline to thinly slice the celery. Slice celery on a diagonal. Combine with celeriac julienne.
In a food processor, combine buttermilk, vegetable oil, lemon juice, salt, and pepper; puree until smooth. Transfer to a small bowl. Add capers, chervil, and tarragon. Stir to combine.
Pour the dressing over the celeriac and celery. Toss to coat evenly. Taste, and adjust seasoning.
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