diff --git a/book/src/background/polynomials.md b/book/src/background/polynomials.md
index 7846233724..a06b0e88e3 100644
--- a/book/src/background/polynomials.md
+++ b/book/src/background/polynomials.md
@@ -242,7 +242,7 @@ Now, we can write our polynomial as a linear combination of Lagrange basis funct
 
 $$A(X) = \sum_{i = 0}^{n-1} a_i\mathcal{L_i}(X), X \in \mathcal{H},$$
 
-which is equivalent to saying that $p(X)$ evaluates to $a_0$ at $\omega^0$,
+which is equivalent to saying that $A(X)$ evaluates to $a_0$ at $\omega^0$,
 to $a_1$ at $\omega^1$, to $a_2$ at $\omega^2, \cdots,$ and so on.
 
 When working over a multiplicative subgroup, the Lagrange basis function has a convenient
diff --git a/book/src/design/proving-system/circuit-commitments.md b/book/src/design/proving-system/circuit-commitments.md
index 7352788929..ee3d3f3351 100644
--- a/book/src/design/proving-system/circuit-commitments.md
+++ b/book/src/design/proving-system/circuit-commitments.md
@@ -59,7 +59,7 @@ arguments are independent.)
 
 Let $c$ be the number of columns that are enabled for equality constraints.
 
-Let $m$ be the maximum number of columns that can accommodated by a
+Let $m$ be the maximum number of columns that can be accommodated by a
 [column set](permutation.md#spanning-a-large-number-of-columns) without exceeding
 the PLONK configuration's maximum constraint degree.
 
diff --git a/book/src/design/proving-system/vanishing.md b/book/src/design/proving-system/vanishing.md
index 75edd53100..d7e337398d 100644
--- a/book/src/design/proving-system/vanishing.md
+++ b/book/src/design/proving-system/vanishing.md
@@ -55,7 +55,7 @@ $$\mathbf{H} = [\text{Commit}(h_0(X)), \text{Commit}(h_1(X)), \dots, \text{Commi
 
 ## Evaluating the polynomials
 
-At this point, all properties of the circuit have been committed to. The verifier now
+At this point, we have committed to all properties of the circuit. The verifier now
 wants to see if the prover committed to the correct $h(X)$ polynomial. The verifier
 samples $x$, and the prover produces the purported evaluations of the various polynomials
 at $x$, for all the relative offsets used in the circuit, as well as $h(X)$.